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bsc-leon-vatthauer/agda/bsc-thesis/Categories.Object.Exponential.html
Leon Vatthauer 2bf4655eae
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<html><head><meta charset="utf-8"><title>Categories.Object.Exponential</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Symbol">{-#</a> <a id="5" class="Keyword">OPTIONS</a> <a id="13" class="Pragma">--without-K</a> <a id="25" class="Pragma">--safe</a> <a id="32" class="Symbol">#-}</a>
<a id="36" class="Keyword">open</a> <a id="41" class="Keyword">import</a> <a id="48" href="Categories.Category.html" class="Module">Categories.Category</a>
<a id="69" class="Comment">-- Exponential Object</a>
<a id="92" class="Comment">-- TODO: Where is the notation from? It is neither from Wikipedia nor the nLab.</a>
<a id="172" class="Keyword">module</a> <a id="179" href="Categories.Object.Exponential.html" class="Module">Categories.Object.Exponential</a> <a id="209" class="Symbol">{</a><a id="210" href="Categories.Object.Exponential.html#210" class="Bound">o</a> <a id="212" href="Categories.Object.Exponential.html#212" class="Bound">ℓ</a> <a id="214" href="Categories.Object.Exponential.html#214" class="Bound">e</a><a id="215" class="Symbol">}</a> <a id="217" class="Symbol">(</a><a id="218" href="Categories.Object.Exponential.html#218" class="Bound">𝒞</a> <a id="220" class="Symbol">:</a> <a id="222" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="231" href="Categories.Object.Exponential.html#210" class="Bound">o</a> <a id="233" href="Categories.Object.Exponential.html#212" class="Bound">ℓ</a> <a id="235" href="Categories.Object.Exponential.html#214" class="Bound">e</a><a id="236" class="Symbol">)</a> <a id="238" class="Keyword">where</a>
<a id="245" class="Keyword">open</a> <a id="250" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="259" href="Categories.Object.Exponential.html#218" class="Bound">𝒞</a>
<a id="262" class="Keyword">open</a> <a id="267" class="Keyword">import</a> <a id="274" href="Level.html" class="Module">Level</a>
<a id="280" class="Keyword">open</a> <a id="285" class="Keyword">import</a> <a id="292" href="Function.html" class="Module">Function</a> <a id="301" class="Keyword">using</a> <a id="307" class="Symbol">(</a><a id="308" href="Function.Base.html#1974" class="Function Operator">_$_</a><a id="311" class="Symbol">)</a>
<a id="314" class="Keyword">open</a> <a id="319" class="Keyword">import</a> <a id="326" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="356" href="Categories.Object.Exponential.html#218" class="Bound">𝒞</a>
<a id="358" class="Keyword">open</a> <a id="363" class="Keyword">import</a> <a id="370" href="Categories.Object.Product.html" class="Module">Categories.Object.Product</a> <a id="396" href="Categories.Object.Exponential.html#218" class="Bound">𝒞</a>
<a id="400" class="Keyword">hiding</a> <a id="407" class="Symbol">(</a><a id="408" href="Categories.Object.Product.Core.html#2078" class="Function">repack</a><a id="414" class="Symbol">;</a> <a id="416" href="Categories.Object.Product.Core.html#2384" class="Function">repack≡id</a><a id="425" class="Symbol">;</a> <a id="427" href="Categories.Object.Product.Core.html#2168" class="Function">repack∘</a><a id="434" class="Symbol">;</a> <a id="436" href="Categories.Object.Product.Core.html#2451" class="Function">repack-cancel</a><a id="449" class="Symbol">;</a> <a id="451" href="Categories.Object.Product.Core.html#2581" class="Function">up-to-iso</a><a id="460" class="Symbol">;</a> <a id="462" href="Categories.Object.Product.Core.html#2819" class="Function">transport-by-iso</a><a id="478" class="Symbol">)</a>
<a id="480" class="Keyword">open</a> <a id="485" class="Keyword">import</a> <a id="492" href="Categories.Morphism.html" class="Module">Categories.Morphism</a> <a id="512" href="Categories.Object.Exponential.html#218" class="Bound">𝒞</a>
<a id="515" class="Keyword">open</a> <a id="520" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="534" class="Keyword">private</a>
<a id="544" class="Keyword">variable</a>
<a id="557" href="Categories.Object.Exponential.html#557" class="Generalizable">A</a> <a id="559" href="Categories.Object.Exponential.html#559" class="Generalizable">B</a> <a id="561" href="Categories.Object.Exponential.html#561" class="Generalizable">C</a> <a id="563" href="Categories.Object.Exponential.html#563" class="Generalizable">D</a> <a id="565" href="Categories.Object.Exponential.html#565" class="Generalizable">X</a> <a id="567" href="Categories.Object.Exponential.html#567" class="Generalizable">Y</a> <a id="569" class="Symbol">:</a> <a id="571" href="Categories.Category.Core.html#559" class="Field">Obj</a>
<a id="576" class="Keyword">record</a> <a id="Exponential"></a><a id="583" href="Categories.Object.Exponential.html#583" class="Record">Exponential</a> <a id="595" class="Symbol">(</a><a id="596" href="Categories.Object.Exponential.html#596" class="Bound">A</a> <a id="598" href="Categories.Object.Exponential.html#598" class="Bound">B</a> <a id="600" class="Symbol">:</a> <a id="602" href="Categories.Category.Core.html#559" class="Field">Obj</a><a id="605" class="Symbol">)</a> <a id="607" class="Symbol">:</a> <a id="609" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="613" class="Symbol">(</a><a id="614" href="Categories.Object.Exponential.html#210" class="Bound">o</a> <a id="616" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="618" href="Categories.Object.Exponential.html#212" class="Bound">ℓ</a> <a id="620" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="622" href="Categories.Object.Exponential.html#214" class="Bound">e</a><a id="623" class="Symbol">)</a> <a id="625" class="Keyword">where</a>
<a id="633" class="Keyword">field</a>
<a id="Exponential.B^A"></a><a id="643" href="Categories.Object.Exponential.html#643" class="Field">B^A</a> <a id="647" class="Symbol">:</a> <a id="649" href="Categories.Category.Core.html#559" class="Field">Obj</a>
<a id="Exponential.product"></a><a id="657" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="665" class="Symbol">:</a> <a id="667" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="675" href="Categories.Object.Exponential.html#643" class="Field">B^A</a> <a id="679" href="Categories.Object.Exponential.html#596" class="Bound">A</a>
<a id="684" class="Keyword">module</a> <a id="Exponential.product"></a><a id="691" href="Categories.Object.Exponential.html#691" class="Module">product</a> <a id="699" class="Symbol">=</a> <a id="701" href="Categories.Object.Product.Core.html#435" class="Module">Product</a> <a id="709" href="Categories.Object.Exponential.html#657" class="Field">product</a>
<a id="Exponential.B^A×A"></a><a id="720" href="Categories.Object.Exponential.html#720" class="Function">B^A×A</a> <a id="726" class="Symbol">:</a> <a id="728" href="Categories.Category.Core.html#559" class="Field">Obj</a>
<a id="734" href="Categories.Object.Exponential.html#720" class="Function">B^A×A</a> <a id="740" class="Symbol">=</a> <a id="742" href="Categories.Object.Product.Core.html#510" class="Function">product.A×B</a>
<a id="757" class="Keyword">field</a>
<a id="Exponential.eval"></a><a id="767" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="776" class="Symbol">:</a> <a id="778" href="Categories.Object.Exponential.html#720" class="Function">B^A×A</a> <a id="784" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="786" href="Categories.Object.Exponential.html#598" class="Bound">B</a>
<a id="Exponential.λg"></a><a id="792" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="801" class="Symbol">:</a> <a id="803" class="Symbol">∀</a> <a id="805" class="Symbol">(</a><a id="806" href="Categories.Object.Exponential.html#806" class="Bound">X×A</a> <a id="810" class="Symbol">:</a> <a id="812" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="820" href="Categories.Object.Exponential.html#565" class="Generalizable">X</a> <a id="822" href="Categories.Object.Exponential.html#596" class="Bound">A</a><a id="823" class="Symbol">)</a> <a id="825" class="Symbol">→</a> <a id="827" class="Symbol">(</a><a id="828" href="Categories.Object.Product.Core.html#510" class="Field">Product.A×B</a> <a id="840" href="Categories.Object.Exponential.html#806" class="Bound">X×A</a> <a id="844" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="846" href="Categories.Object.Exponential.html#598" class="Bound">B</a><a id="847" class="Symbol">)</a> <a id="849" class="Symbol">→</a> <a id="851" class="Symbol">(</a><a id="852" href="Categories.Object.Exponential.html#565" class="Generalizable">X</a> <a id="854" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="856" href="Categories.Object.Exponential.html#643" class="Field">B^A</a><a id="859" class="Symbol">)</a>
<a id="Exponential.β"></a><a id="865" href="Categories.Object.Exponential.html#865" class="Field">β</a> <a id="874" class="Symbol">:</a> <a id="876" class="Symbol">∀</a> <a id="878" class="Symbol">(</a><a id="879" href="Categories.Object.Exponential.html#879" class="Bound">X×A</a> <a id="883" class="Symbol">:</a> <a id="885" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="893" href="Categories.Object.Exponential.html#565" class="Generalizable">X</a> <a id="895" href="Categories.Object.Exponential.html#596" class="Bound">A</a><a id="896" class="Symbol">)</a> <a id="898" class="Symbol">{</a><a id="899" href="Categories.Object.Exponential.html#899" class="Bound">g</a> <a id="901" class="Symbol">:</a> <a id="903" href="Categories.Object.Product.Core.html#510" class="Field">Product.A×B</a> <a id="915" href="Categories.Object.Exponential.html#879" class="Bound">X×A</a> <a id="919" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="921" href="Categories.Object.Exponential.html#598" class="Bound">B</a><a id="922" class="Symbol">}</a> <a id="924" class="Symbol">→</a>
<a id="943" class="Symbol">(</a><a id="944" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="949" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="951" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="953" href="Categories.Object.Exponential.html#879" class="Bound">X×A</a> <a id="957" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="959" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="967" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="969" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="972" href="Categories.Object.Exponential.html#879" class="Bound">X×A</a> <a id="976" href="Categories.Object.Exponential.html#899" class="Bound">g</a> <a id="978" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a> <a id="982" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="984" href="Categories.Object.Exponential.html#899" class="Bound">g</a><a id="985" class="Symbol">)</a>
<a id="Exponential.λ-unique"></a><a id="991" href="Categories.Object.Exponential.html#991" class="Field">λ-unique</a> <a id="1000" class="Symbol">:</a> <a id="1002" class="Symbol">∀</a> <a id="1004" class="Symbol">(</a><a id="1005" href="Categories.Object.Exponential.html#1005" class="Bound">X×A</a> <a id="1009" class="Symbol">:</a> <a id="1011" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="1019" href="Categories.Object.Exponential.html#565" class="Generalizable">X</a> <a id="1021" href="Categories.Object.Exponential.html#596" class="Bound">A</a><a id="1022" class="Symbol">)</a> <a id="1024" class="Symbol">{</a><a id="1025" href="Categories.Object.Exponential.html#1025" class="Bound">g</a> <a id="1027" class="Symbol">:</a> <a id="1029" href="Categories.Object.Product.Core.html#510" class="Field">Product.A×B</a> <a id="1041" href="Categories.Object.Exponential.html#1005" class="Bound">X×A</a> <a id="1045" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="1047" href="Categories.Object.Exponential.html#598" class="Bound">B</a><a id="1048" class="Symbol">}</a> <a id="1050" class="Symbol">{</a><a id="1051" href="Categories.Object.Exponential.html#1051" class="Bound">h</a> <a id="1053" class="Symbol">:</a> <a id="1055" href="Categories.Object.Exponential.html#565" class="Generalizable">X</a> <a id="1057" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="1059" href="Categories.Object.Exponential.html#643" class="Field">B^A</a><a id="1062" class="Symbol">}</a> <a id="1064" class="Symbol">→</a>
<a id="1083" class="Symbol">(</a><a id="1084" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="1089" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1091" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="1093" href="Categories.Object.Exponential.html#1005" class="Bound">X×A</a> <a id="1097" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="1099" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="1107" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="1109" href="Categories.Object.Exponential.html#1051" class="Bound">h</a> <a id="1111" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a> <a id="1115" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1117" href="Categories.Object.Exponential.html#1025" class="Bound">g</a><a id="1118" class="Symbol">)</a> <a id="1120" class="Symbol">→</a> <a id="1122" class="Symbol">(</a><a id="1123" href="Categories.Object.Exponential.html#1051" class="Bound">h</a> <a id="1125" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1127" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="1130" href="Categories.Object.Exponential.html#1005" class="Bound">X×A</a> <a id="1134" href="Categories.Object.Exponential.html#1025" class="Bound">g</a><a id="1135" class="Symbol">)</a>
<a id="Exponential.η"></a><a id="1140" href="Categories.Object.Exponential.html#1140" class="Function">η</a> <a id="1142" class="Symbol">:</a> <a id="1144" class="Symbol">∀</a> <a id="1146" class="Symbol">(</a><a id="1147" href="Categories.Object.Exponential.html#1147" class="Bound">X×A</a> <a id="1151" class="Symbol">:</a> <a id="1153" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="1161" href="Categories.Object.Exponential.html#565" class="Generalizable">X</a> <a id="1163" href="Categories.Object.Exponential.html#596" class="Bound">A</a><a id="1164" class="Symbol">)</a> <a id="1166" class="Symbol">{</a><a id="1167" href="Categories.Object.Exponential.html#1167" class="Bound">f</a> <a id="1169" class="Symbol">:</a> <a id="1171" href="Categories.Object.Exponential.html#565" class="Generalizable">X</a> <a id="1173" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="1175" href="Categories.Object.Exponential.html#643" class="Field">B^A</a> <a id="1179" class="Symbol">}</a> <a id="1181" class="Symbol">→</a> <a id="1183" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="1186" href="Categories.Object.Exponential.html#1147" class="Bound">X×A</a> <a id="1190" class="Symbol">(</a><a id="1191" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="1196" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1198" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="1200" href="Categories.Object.Exponential.html#1147" class="Bound">X×A</a> <a id="1204" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="1206" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="1214" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="1216" href="Categories.Object.Exponential.html#1167" class="Bound">f</a> <a id="1218" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a><a id="1221" class="Symbol">)</a> <a id="1223" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1225" href="Categories.Object.Exponential.html#1167" class="Bound">f</a>
<a id="1229" href="Categories.Object.Exponential.html#1140" class="Function">η</a> <a id="1231" href="Categories.Object.Exponential.html#1231" class="Bound">X×A</a> <a id="1235" class="Symbol">{</a><a id="1236" href="Categories.Object.Exponential.html#1236" class="Bound">f</a><a id="1237" class="Symbol">}</a> <a id="1239" class="Symbol">=</a> <a id="1241" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="1243" class="Symbol">(</a><a id="1244" href="Categories.Object.Exponential.html#991" class="Field">λ-unique</a> <a id="1253" href="Categories.Object.Exponential.html#1231" class="Bound">X×A</a> <a id="1257" href="Relation.Binary.Structures.html#1596" class="Function">Equiv.refl</a><a id="1267" class="Symbol">)</a>
<a id="Exponential.λ-cong"></a><a id="1272" href="Categories.Object.Exponential.html#1272" class="Function">λ-cong</a> <a id="1279" class="Symbol">:</a> <a id="1281" class="Symbol">∀</a> <a id="1283" class="Symbol">{</a><a id="1284" href="Categories.Object.Exponential.html#1284" class="Bound">X</a> <a id="1286" class="Symbol">:</a> <a id="1288" href="Categories.Category.Core.html#559" class="Field">Obj</a><a id="1291" class="Symbol">}</a> <a id="1293" class="Symbol">(</a><a id="1294" href="Categories.Object.Exponential.html#1294" class="Bound">X×A</a> <a id="1298" class="Symbol">:</a> <a id="1300" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="1308" href="Categories.Object.Exponential.html#1284" class="Bound">X</a> <a id="1310" href="Categories.Object.Exponential.html#596" class="Bound">A</a><a id="1311" class="Symbol">)</a> <a id="1313" class="Symbol">{</a><a id="1314" href="Categories.Object.Exponential.html#1314" class="Bound">f</a> <a id="1316" href="Categories.Object.Exponential.html#1316" class="Bound">g</a><a id="1317" class="Symbol">}</a> <a id="1319" class="Symbol">→</a>
<a id="1334" href="Categories.Object.Exponential.html#1314" class="Bound">f</a> <a id="1336" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1338" href="Categories.Object.Exponential.html#1316" class="Bound">g</a> <a id="1340" class="Symbol">→</a> <a id="1342" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="1345" href="Categories.Object.Exponential.html#1294" class="Bound">X×A</a> <a id="1349" href="Categories.Object.Exponential.html#1314" class="Bound">f</a> <a id="1351" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1353" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="1356" href="Categories.Object.Exponential.html#1294" class="Bound">X×A</a> <a id="1360" href="Categories.Object.Exponential.html#1316" class="Bound">g</a>
<a id="1364" href="Categories.Object.Exponential.html#1272" class="Function">λ-cong</a> <a id="1371" href="Categories.Object.Exponential.html#1371" class="Bound">X×A</a> <a id="1375" class="Symbol">{</a><a id="1376" class="Argument">f</a> <a id="1378" class="Symbol">=</a> <a id="1380" href="Categories.Object.Exponential.html#1380" class="Bound">f</a><a id="1381" class="Symbol">}</a> <a id="1383" class="Symbol">{</a><a id="1384" class="Argument">g</a> <a id="1386" class="Symbol">=</a> <a id="1388" href="Categories.Object.Exponential.html#1388" class="Bound">g</a><a id="1389" class="Symbol">}</a> <a id="1391" href="Categories.Object.Exponential.html#1391" class="Bound">f≡g</a> <a id="1395" class="Symbol">=</a> <a id="1397" href="Categories.Object.Exponential.html#991" class="Field">λ-unique</a> <a id="1406" href="Categories.Object.Exponential.html#1371" class="Bound">X×A</a> <a id="1410" class="Symbol">(</a><a id="1411" href="Categories.Object.Exponential.html#865" class="Field">β</a> <a id="1413" href="Categories.Object.Exponential.html#1371" class="Bound">X×A</a> <a id="1417" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1419" href="Categories.Object.Exponential.html#1391" class="Bound">f≡g</a><a id="1422" class="Symbol">)</a>
<a id="Exponential.λ-inj"></a><a id="1427" href="Categories.Object.Exponential.html#1427" class="Function">λ-inj</a> <a id="1433" class="Symbol">:</a> <a id="1435" class="Symbol">∀</a> <a id="1437" class="Symbol">{</a><a id="1438" href="Categories.Object.Exponential.html#1438" class="Bound">X</a> <a id="1440" class="Symbol">:</a> <a id="1442" href="Categories.Category.Core.html#559" class="Field">Obj</a><a id="1445" class="Symbol">}</a> <a id="1447" class="Symbol">(</a><a id="1448" href="Categories.Object.Exponential.html#1448" class="Bound">X×A</a> <a id="1452" class="Symbol">:</a> <a id="1454" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="1462" href="Categories.Object.Exponential.html#1438" class="Bound">X</a> <a id="1464" href="Categories.Object.Exponential.html#596" class="Bound">A</a><a id="1465" class="Symbol">)</a> <a id="1467" class="Symbol">{</a><a id="1468" href="Categories.Object.Exponential.html#1468" class="Bound">f</a> <a id="1470" href="Categories.Object.Exponential.html#1470" class="Bound">g</a><a id="1471" class="Symbol">}</a> <a id="1473" class="Symbol">→</a> <a id="1475" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="1478" href="Categories.Object.Exponential.html#1448" class="Bound">X×A</a> <a id="1482" href="Categories.Object.Exponential.html#1468" class="Bound">f</a> <a id="1484" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1486" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="1489" href="Categories.Object.Exponential.html#1448" class="Bound">X×A</a> <a id="1493" href="Categories.Object.Exponential.html#1470" class="Bound">g</a> <a id="1495" class="Symbol">→</a> <a id="1497" href="Categories.Object.Exponential.html#1468" class="Bound">f</a> <a id="1499" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1501" href="Categories.Object.Exponential.html#1470" class="Bound">g</a>
<a id="1505" href="Categories.Object.Exponential.html#1427" class="Function">λ-inj</a> <a id="1511" href="Categories.Object.Exponential.html#1511" class="Bound">X×A</a> <a id="1515" class="Symbol">{</a><a id="1516" class="Argument">f</a> <a id="1518" class="Symbol">=</a> <a id="1520" href="Categories.Object.Exponential.html#1520" class="Bound">f</a><a id="1521" class="Symbol">}</a> <a id="1523" class="Symbol">{</a><a id="1524" class="Argument">g</a> <a id="1526" class="Symbol">=</a> <a id="1528" href="Categories.Object.Exponential.html#1528" class="Bound">g</a><a id="1529" class="Symbol">}</a> <a id="1531" href="Categories.Object.Exponential.html#1531" class="Bound">eq</a> <a id="1534" class="Symbol">=</a> <a id="1536" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="1546" href="Categories.Object.Exponential.html#1520" class="Bound">f</a> <a id="1585" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="1589" href="Categories.Object.Exponential.html#865" class="Field">β</a> <a id="1591" href="Categories.Object.Exponential.html#1511" class="Bound">X×A</a> <a id="1595" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="1601" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="1606" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1608" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="1610" href="Categories.Object.Exponential.html#1511" class="Bound">X×A</a> <a id="1614" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="1616" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="1624" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="1626" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="1629" href="Categories.Object.Exponential.html#1511" class="Bound">X×A</a> <a id="1633" href="Categories.Object.Exponential.html#1520" class="Bound">f</a> <a id="1635" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a> <a id="1640" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1643" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="1651" href="Categories.Object.Product.Morphisms.html#1731" class="Function Operator">[</a> <a id="1653" href="Categories.Object.Exponential.html#1511" class="Bound">X×A</a> <a id="1657" href="Categories.Object.Product.Morphisms.html#1731" class="Function Operator">⇒</a> <a id="1659" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="1668" href="Categories.Object.Product.Morphisms.html#1731" class="Function Operator">]×-cong₂</a> <a id="1677" href="Categories.Object.Exponential.html#1531" class="Bound">eq</a> <a id="1680" href="Relation.Binary.Structures.html#1596" class="Function">Equiv.refl</a> <a id="1691" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1697" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="1702" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1704" href="Categories.Object.Product.Morphisms.html#838" class="Function Operator">[</a> <a id="1706" href="Categories.Object.Exponential.html#1511" class="Bound">X×A</a> <a id="1710" href="Categories.Object.Product.Morphisms.html#838" class="Function Operator">⇒</a> <a id="1712" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="1720" href="Categories.Object.Product.Morphisms.html#838" class="Function Operator">]</a> <a id="1722" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="1725" href="Categories.Object.Exponential.html#1511" class="Bound">X×A</a> <a id="1729" href="Categories.Object.Exponential.html#1528" class="Bound">g</a> <a id="1731" href="Categories.Object.Product.Morphisms.html#838" class="Function Operator">×</a> <a id="1733" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="1736" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1739" href="Categories.Object.Exponential.html#865" class="Field">β</a> <a id="1741" href="Categories.Object.Exponential.html#1511" class="Bound">X×A</a> <a id="1745" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1751" href="Categories.Object.Exponential.html#1528" class="Bound">g</a> <a id="1790" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="Exponential.subst"></a><a id="1795" href="Categories.Object.Exponential.html#1795" class="Function">subst</a> <a id="1801" class="Symbol">:</a> <a id="1803" class="Symbol">∀</a> <a id="1805" class="Symbol">(</a><a id="1806" href="Categories.Object.Exponential.html#1806" class="Bound">p₂</a> <a id="1809" class="Symbol">:</a> <a id="1811" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="1819" href="Categories.Object.Exponential.html#561" class="Generalizable">C</a> <a id="1821" href="Categories.Object.Exponential.html#596" class="Bound">A</a><a id="1822" class="Symbol">)</a> <a id="1824" class="Symbol">(</a><a id="1825" href="Categories.Object.Exponential.html#1825" class="Bound">p₃</a> <a id="1828" class="Symbol">:</a> <a id="1830" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="1838" href="Categories.Object.Exponential.html#563" class="Generalizable">D</a> <a id="1840" href="Categories.Object.Exponential.html#596" class="Bound">A</a><a id="1841" class="Symbol">)</a> <a id="1843" class="Symbol">{</a><a id="1844" href="Categories.Object.Exponential.html#1844" class="Bound">f</a> <a id="1846" href="Categories.Object.Exponential.html#1846" class="Bound">g</a><a id="1847" class="Symbol">}</a> <a id="1849" class="Symbol">→</a>
<a id="1863" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="1866" href="Categories.Object.Exponential.html#1825" class="Bound">p₃</a> <a id="1869" href="Categories.Object.Exponential.html#1844" class="Bound">f</a> <a id="1871" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1873" href="Categories.Object.Exponential.html#1846" class="Bound">g</a> <a id="1875" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1877" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="1880" href="Categories.Object.Exponential.html#1806" class="Bound">p₂</a> <a id="1883" class="Symbol">(</a><a id="1884" href="Categories.Object.Exponential.html#1844" class="Bound">f</a> <a id="1886" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1888" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="1890" href="Categories.Object.Exponential.html#1806" class="Bound">p₂</a> <a id="1893" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="1895" href="Categories.Object.Exponential.html#1825" class="Bound">p₃</a> <a id="1898" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="1900" href="Categories.Object.Exponential.html#1846" class="Bound">g</a> <a id="1902" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a><a id="1905" class="Symbol">)</a>
<a id="1909" href="Categories.Object.Exponential.html#1795" class="Function">subst</a> <a id="1915" href="Categories.Object.Exponential.html#1915" class="Bound">p₂</a> <a id="1918" href="Categories.Object.Exponential.html#1918" class="Bound">p₃</a> <a id="1921" class="Symbol">{</a><a id="1922" href="Categories.Object.Exponential.html#1922" class="Bound">f</a><a id="1923" class="Symbol">}</a> <a id="1925" class="Symbol">{</a><a id="1926" href="Categories.Object.Exponential.html#1926" class="Bound">g</a><a id="1927" class="Symbol">}</a> <a id="1929" class="Symbol">=</a> <a id="1931" href="Categories.Object.Exponential.html#991" class="Field">λ-unique</a> <a id="1940" href="Categories.Object.Exponential.html#1915" class="Bound">p₂</a> <a id="1943" class="Symbol">(</a><a id="1944" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="1954" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="1959" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1961" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="1963" href="Categories.Object.Exponential.html#1915" class="Bound">p₂</a> <a id="1966" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="1968" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="1976" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="1978" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="1981" href="Categories.Object.Exponential.html#1918" class="Bound">p₃</a> <a id="1984" href="Categories.Object.Exponential.html#1922" class="Bound">f</a> <a id="1986" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1988" href="Categories.Object.Exponential.html#1926" class="Bound">g</a> <a id="1990" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a>
<a id="2021" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="2025" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="2033" href="Categories.Object.Product.Morphisms.html#5474" class="Function Operator">[</a> <a id="2035" href="Categories.Object.Exponential.html#1915" class="Bound">p₂</a> <a id="2038" href="Categories.Object.Product.Morphisms.html#5474" class="Function Operator">⇒</a> <a id="2040" href="Categories.Object.Exponential.html#1918" class="Bound">p₃</a> <a id="2043" href="Categories.Object.Product.Morphisms.html#5474" class="Function Operator">⇒</a> <a id="2045" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="2053" href="Categories.Object.Product.Morphisms.html#5474" class="Function Operator">]×id∘×id</a> <a id="2062" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="2068" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="2073" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2075" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="2077" href="Categories.Object.Exponential.html#1918" class="Bound">p₃</a> <a id="2080" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="2082" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="2090" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="2092" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="2095" href="Categories.Object.Exponential.html#1918" class="Bound">p₃</a> <a id="2098" href="Categories.Object.Exponential.html#1922" class="Bound">f</a> <a id="2100" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a> <a id="2104" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2106" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="2108" href="Categories.Object.Exponential.html#1915" class="Bound">p₂</a> <a id="2111" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="2113" href="Categories.Object.Exponential.html#1918" class="Bound">p₃</a> <a id="2116" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="2118" href="Categories.Object.Exponential.html#1926" class="Bound">g</a> <a id="2120" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a>
<a id="2151" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2154" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="2160" class="Symbol">(</a><a id="2161" href="Categories.Object.Exponential.html#865" class="Field">β</a> <a id="2163" href="Categories.Object.Exponential.html#1918" class="Bound">p₃</a><a id="2165" class="Symbol">)</a> <a id="2167" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2173" href="Categories.Object.Exponential.html#1922" class="Bound">f</a> <a id="2175" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2177" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="2179" href="Categories.Object.Exponential.html#1915" class="Bound">p₂</a> <a id="2182" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="2184" href="Categories.Object.Exponential.html#1918" class="Bound">p₃</a> <a id="2187" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="2189" href="Categories.Object.Exponential.html#1926" class="Bound">g</a> <a id="2191" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a> <a id="2195" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a><a id="2196" class="Symbol">)</a>
<a id="Exponential.η-id"></a><a id="2201" href="Categories.Object.Exponential.html#2201" class="Function">η-id</a> <a id="2206" class="Symbol">:</a> <a id="2208" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="2211" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="2219" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="2224" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="2226" href="Categories.Category.Core.html#630" class="Field">id</a>
<a id="2231" href="Categories.Object.Exponential.html#2201" class="Function">η-id</a> <a id="2236" class="Symbol">=</a> <a id="2238" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="2248" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="2251" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="2259" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="2297" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="2301" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a> <a id="2311" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="2317" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="2320" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="2328" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="2333" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2335" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="2366" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2369" href="Categories.Object.Exponential.html#1795" class="Function">subst</a> <a id="2375" class="Symbol">_</a> <a id="2377" class="Symbol">_</a> <a id="2379" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2385" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="2388" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="2396" class="Symbol">(</a><a id="2397" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="2402" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2404" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="2406" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="2414" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="2416" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="2424" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="2426" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="2429" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a><a id="2432" class="Symbol">)</a> <a id="2434" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2437" href="Categories.Object.Exponential.html#1140" class="Function">η</a> <a id="2439" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="2447" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2453" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="2502" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="Exponential.λ-unique′"></a><a id="2507" href="Categories.Object.Exponential.html#2507" class="Function">λ-unique′</a> <a id="2517" class="Symbol">:</a> <a id="2519" class="Symbol">∀</a> <a id="2521" class="Symbol">(</a><a id="2522" href="Categories.Object.Exponential.html#2522" class="Bound">X×A</a> <a id="2526" class="Symbol">:</a> <a id="2528" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="2536" href="Categories.Object.Exponential.html#565" class="Generalizable">X</a> <a id="2538" href="Categories.Object.Exponential.html#596" class="Bound">A</a><a id="2539" class="Symbol">)</a> <a id="2541" class="Symbol">{</a><a id="2542" href="Categories.Object.Exponential.html#2542" class="Bound">h</a> <a id="2544" href="Categories.Object.Exponential.html#2544" class="Bound">i</a> <a id="2546" class="Symbol">:</a> <a id="2548" href="Categories.Object.Exponential.html#565" class="Generalizable">X</a> <a id="2550" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="2552" href="Categories.Object.Exponential.html#643" class="Field">B^A</a><a id="2555" class="Symbol">}</a> <a id="2557" class="Symbol">→</a>
<a id="2575" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="2580" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2582" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="2584" href="Categories.Object.Exponential.html#2522" class="Bound">X×A</a> <a id="2588" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="2590" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="2598" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="2600" href="Categories.Object.Exponential.html#2542" class="Bound">h</a> <a id="2602" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a> <a id="2606" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="2608" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="2613" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2615" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="2617" href="Categories.Object.Exponential.html#2522" class="Bound">X×A</a> <a id="2621" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="2623" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="2631" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="2633" href="Categories.Object.Exponential.html#2544" class="Bound">i</a> <a id="2635" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a> <a id="2639" class="Symbol">→</a> <a id="2641" href="Categories.Object.Exponential.html#2542" class="Bound">h</a> <a id="2643" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="2645" href="Categories.Object.Exponential.html#2544" class="Bound">i</a>
<a id="2649" href="Categories.Object.Exponential.html#2507" class="Function">λ-unique′</a> <a id="2659" href="Categories.Object.Exponential.html#2659" class="Bound">p</a> <a id="2661" href="Categories.Object.Exponential.html#2661" class="Bound">eq</a> <a id="2664" class="Symbol">=</a> <a id="2666" href="Categories.Object.Exponential.html#991" class="Field">λ-unique</a> <a id="2675" href="Categories.Object.Exponential.html#2659" class="Bound">p</a> <a id="2677" href="Categories.Object.Exponential.html#2661" class="Bound">eq</a> <a id="2680" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2682" class="Symbol">(</a><a id="2683" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="2685" class="Symbol">(</a><a id="2686" href="Categories.Object.Exponential.html#991" class="Field">λ-unique</a> <a id="2695" href="Categories.Object.Exponential.html#2659" class="Bound">p</a> <a id="2697" href="Relation.Binary.Structures.html#1596" class="Function">Equiv.refl</a><a id="2707" class="Symbol">))</a>
<a id="2711" class="Comment">-- some aliases to make proof signatures less ugly</a>
<a id="[_]eval"></a><a id="2762" href="Categories.Object.Exponential.html#2762" class="Function Operator">[_]eval</a> <a id="2770" class="Symbol">:</a> <a id="2772" class="Symbol">∀{</a><a id="2774" href="Categories.Object.Exponential.html#2774" class="Bound">A</a> <a id="2776" href="Categories.Object.Exponential.html#2776" class="Bound">B</a><a id="2777" class="Symbol">}(</a><a id="2779" href="Categories.Object.Exponential.html#2779" class="Bound">e₁</a> <a id="2782" class="Symbol">:</a> <a id="2784" href="Categories.Object.Exponential.html#583" class="Record">Exponential</a> <a id="2796" href="Categories.Object.Exponential.html#2774" class="Bound">A</a> <a id="2798" href="Categories.Object.Exponential.html#2776" class="Bound">B</a><a id="2799" class="Symbol">)</a> <a id="2801" class="Symbol">→</a> <a id="2803" href="Categories.Object.Exponential.html#720" class="Function">Exponential.B^A×A</a> <a id="2821" href="Categories.Object.Exponential.html#2779" class="Bound">e₁</a> <a id="2824" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="2826" href="Categories.Object.Exponential.html#2776" class="Bound">B</a>
<a id="2828" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="2830" href="Categories.Object.Exponential.html#2830" class="Bound">e₁</a> <a id="2833" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a> <a id="2839" class="Symbol">=</a> <a id="2841" href="Categories.Object.Exponential.html#767" class="Field">Exponential.eval</a> <a id="2858" href="Categories.Object.Exponential.html#2830" class="Bound">e₁</a>
<a id="[_]λ"></a><a id="2862" href="Categories.Object.Exponential.html#2862" class="Function Operator">[_]λ</a> <a id="2867" class="Symbol">:</a> <a id="2869" class="Symbol">∀{</a><a id="2871" href="Categories.Object.Exponential.html#2871" class="Bound">A</a> <a id="2873" href="Categories.Object.Exponential.html#2873" class="Bound">B</a><a id="2874" class="Symbol">}(</a><a id="2876" href="Categories.Object.Exponential.html#2876" class="Bound">e₁</a> <a id="2879" class="Symbol">:</a> <a id="2881" href="Categories.Object.Exponential.html#583" class="Record">Exponential</a> <a id="2893" href="Categories.Object.Exponential.html#2871" class="Bound">A</a> <a id="2895" href="Categories.Object.Exponential.html#2873" class="Bound">B</a><a id="2896" class="Symbol">)</a>
<a id="2900" class="Symbol">→</a> <a id="2902" class="Symbol">{</a><a id="2903" href="Categories.Object.Exponential.html#2903" class="Bound">X</a> <a id="2905" class="Symbol">:</a> <a id="2907" href="Categories.Category.Core.html#559" class="Field">Obj</a><a id="2910" class="Symbol">}</a> <a id="2912" class="Symbol">→</a> <a id="2914" class="Symbol">(</a><a id="2915" href="Categories.Object.Exponential.html#2915" class="Bound">X×A</a> <a id="2919" class="Symbol">:</a> <a id="2921" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="2929" href="Categories.Object.Exponential.html#2903" class="Bound">X</a> <a id="2931" href="Categories.Object.Exponential.html#2871" class="Bound">A</a><a id="2932" class="Symbol">)</a> <a id="2934" class="Symbol">→</a> <a id="2936" class="Symbol">(</a><a id="2937" href="Categories.Object.Product.Core.html#510" class="Field">Product.A×B</a> <a id="2949" href="Categories.Object.Exponential.html#2915" class="Bound">X×A</a> <a id="2953" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="2955" href="Categories.Object.Exponential.html#2873" class="Bound">B</a><a id="2956" class="Symbol">)</a> <a id="2958" class="Symbol">→</a> <a id="2960" class="Symbol">(</a><a id="2961" href="Categories.Object.Exponential.html#2903" class="Bound">X</a> <a id="2963" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="2965" href="Categories.Object.Exponential.html#643" class="Field">Exponential.B^A</a> <a id="2981" href="Categories.Object.Exponential.html#2876" class="Bound">e₁</a><a id="2983" class="Symbol">)</a>
<a id="2985" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="2987" href="Categories.Object.Exponential.html#2987" class="Bound">e₁</a> <a id="2990" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="2993" class="Symbol">=</a> <a id="2995" href="Categories.Object.Exponential.html#792" class="Field">Exponential.λg</a> <a id="3010" href="Categories.Object.Exponential.html#2987" class="Bound">e₁</a>
<a id="3014" class="Comment">{-
D×C --id × f--&gt; D×A --g--&gt; B
D --λ (g ∘ id × f)--&gt; B^C
\ ^
\ /
λ g λ (e ∘ id × f)
\ /
v /
B^A
-}</a>
<a id="λ-distrib"></a><a id="3212" href="Categories.Object.Exponential.html#3212" class="Function">λ-distrib</a> <a id="3222" class="Symbol">:</a> <a id="3224" class="Symbol">∀</a> <a id="3226" class="Symbol">(</a><a id="3227" href="Categories.Object.Exponential.html#3227" class="Bound">e₁</a> <a id="3230" class="Symbol">:</a> <a id="3232" href="Categories.Object.Exponential.html#583" class="Record">Exponential</a> <a id="3244" href="Categories.Object.Exponential.html#561" class="Generalizable">C</a> <a id="3246" href="Categories.Object.Exponential.html#559" class="Generalizable">B</a><a id="3247" class="Symbol">)</a> <a id="3249" class="Symbol">(</a><a id="3250" href="Categories.Object.Exponential.html#3250" class="Bound">e₂</a> <a id="3253" class="Symbol">:</a> <a id="3255" href="Categories.Object.Exponential.html#583" class="Record">Exponential</a> <a id="3267" href="Categories.Object.Exponential.html#557" class="Generalizable">A</a> <a id="3269" href="Categories.Object.Exponential.html#559" class="Generalizable">B</a><a id="3270" class="Symbol">)</a>
<a id="3286" class="Symbol">(</a><a id="3287" href="Categories.Object.Exponential.html#3287" class="Bound">p₃</a> <a id="3290" class="Symbol">:</a> <a id="3292" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="3300" href="Categories.Object.Exponential.html#563" class="Generalizable">D</a> <a id="3302" href="Categories.Object.Exponential.html#561" class="Generalizable">C</a><a id="3303" class="Symbol">)</a> <a id="3305" class="Symbol">(</a><a id="3306" href="Categories.Object.Exponential.html#3306" class="Bound">p₄</a> <a id="3309" class="Symbol">:</a> <a id="3311" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="3319" href="Categories.Object.Exponential.html#563" class="Generalizable">D</a> <a id="3321" href="Categories.Object.Exponential.html#557" class="Generalizable">A</a><a id="3322" class="Symbol">)</a> <a id="3324" class="Symbol">(</a><a id="3325" href="Categories.Object.Exponential.html#3325" class="Bound">p₅</a> <a id="3328" class="Symbol">:</a> <a id="3330" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="3338" class="Symbol">(</a><a id="3339" href="Categories.Object.Exponential.html#643" class="Field">Exponential.B^A</a> <a id="3355" href="Categories.Object.Exponential.html#3250" class="Bound">e₂</a><a id="3357" class="Symbol">)</a> <a id="3359" href="Categories.Object.Exponential.html#561" class="Generalizable">C</a><a id="3360" class="Symbol">)</a>
<a id="3376" class="Symbol">{</a><a id="3377" href="Categories.Object.Exponential.html#3377" class="Bound">f</a><a id="3378" class="Symbol">}</a> <a id="3380" class="Symbol">{</a><a id="3381" href="Categories.Object.Exponential.html#3381" class="Bound">g</a> <a id="3383" class="Symbol">:</a> <a id="3385" href="Categories.Object.Product.Core.html#510" class="Field">Product.A×B</a> <a id="3397" href="Categories.Object.Exponential.html#3306" class="Bound">p₄</a> <a id="3400" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="3402" href="Categories.Object.Exponential.html#559" class="Generalizable">B</a><a id="3403" class="Symbol">}</a> <a id="3405" class="Symbol">→</a>
<a id="3421" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="3423" href="Categories.Object.Exponential.html#3227" class="Bound">e₁</a> <a id="3426" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="3429" href="Categories.Object.Exponential.html#3287" class="Bound">p₃</a> <a id="3432" class="Symbol">(</a><a id="3433" href="Categories.Object.Exponential.html#3381" class="Bound">g</a> <a id="3435" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3437" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">[</a> <a id="3439" href="Categories.Object.Exponential.html#3287" class="Bound">p₃</a> <a id="3442" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">⇒</a> <a id="3444" href="Categories.Object.Exponential.html#3306" class="Bound">p₄</a> <a id="3447" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">]id×</a> <a id="3452" href="Categories.Object.Exponential.html#3377" class="Bound">f</a><a id="3453" class="Symbol">)</a>
<a id="3469" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="3471" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="3473" href="Categories.Object.Exponential.html#3227" class="Bound">e₁</a> <a id="3476" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="3479" href="Categories.Object.Exponential.html#3325" class="Bound">p₅</a> <a id="3482" class="Symbol">(</a><a id="3483" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="3485" href="Categories.Object.Exponential.html#3250" class="Bound">e₂</a> <a id="3488" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a> <a id="3494" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3496" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">[</a> <a id="3498" href="Categories.Object.Exponential.html#3325" class="Bound">p₅</a> <a id="3501" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">⇒</a> <a id="3503" href="Categories.Object.Exponential.html#657" class="Field">Exponential.product</a> <a id="3523" href="Categories.Object.Exponential.html#3250" class="Bound">e₂</a> <a id="3526" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">]id×</a> <a id="3531" href="Categories.Object.Exponential.html#3377" class="Bound">f</a><a id="3532" class="Symbol">)</a> <a id="3534" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3536" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="3538" href="Categories.Object.Exponential.html#3250" class="Bound">e₂</a> <a id="3541" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="3544" href="Categories.Object.Exponential.html#3306" class="Bound">p₄</a> <a id="3547" href="Categories.Object.Exponential.html#3381" class="Bound">g</a>
<a id="3549" href="Categories.Object.Exponential.html#3212" class="Function">λ-distrib</a> <a id="3559" href="Categories.Object.Exponential.html#3559" class="Bound">e₁</a> <a id="3562" href="Categories.Object.Exponential.html#3562" class="Bound">e₂</a> <a id="3565" href="Categories.Object.Exponential.html#3565" class="Bound">p₃</a> <a id="3568" href="Categories.Object.Exponential.html#3568" class="Bound">p₄</a> <a id="3571" href="Categories.Object.Exponential.html#3571" class="Bound">p₅</a> <a id="3574" class="Symbol">{</a><a id="3575" href="Categories.Object.Exponential.html#3575" class="Bound">f</a><a id="3576" class="Symbol">}</a> <a id="3578" class="Symbol">{</a><a id="3579" href="Categories.Object.Exponential.html#3579" class="Bound">g</a><a id="3580" class="Symbol">}</a> <a id="3582" class="Symbol">=</a> <a id="3584" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="3586" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="3588" href="Categories.Object.Exponential.html#991" class="Function">e₁.λ-unique</a> <a id="3600" href="Categories.Object.Exponential.html#3565" class="Bound">p₃</a> <a id="3603" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="3605" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="3613" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="3615" href="Categories.Object.Exponential.html#3559" class="Bound">e₁</a> <a id="3618" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a> <a id="3624" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3626" class="Symbol">(</a><a id="3627" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="3629" href="Categories.Object.Exponential.html#3565" class="Bound">p₃</a> <a id="3632" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="3634" href="Categories.Object.Exponential.html#4395" class="Function">p₁</a> <a id="3637" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="3639" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="3641" href="Categories.Object.Exponential.html#3559" class="Bound">e₁</a> <a id="3644" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="3647" href="Categories.Object.Exponential.html#3571" class="Bound">p₅</a> <a id="3650" class="Symbol">(</a><a id="3651" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="3653" href="Categories.Object.Exponential.html#3562" class="Bound">e₂</a> <a id="3656" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a> <a id="3662" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3664" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">[</a> <a id="3666" href="Categories.Object.Exponential.html#3571" class="Bound">p₅</a> <a id="3669" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">⇒</a> <a id="3671" href="Categories.Object.Exponential.html#4426" class="Function">p₂</a> <a id="3674" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">]id×</a> <a id="3679" href="Categories.Object.Exponential.html#3575" class="Bound">f</a><a id="3680" class="Symbol">)</a> <a id="3682" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3684" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="3686" href="Categories.Object.Exponential.html#3562" class="Bound">e₂</a> <a id="3689" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="3692" href="Categories.Object.Exponential.html#3568" class="Bound">p₄</a> <a id="3695" href="Categories.Object.Exponential.html#3579" class="Bound">g</a> <a id="3697" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a><a id="3700" class="Symbol">)</a>
<a id="3725" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="3729" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="3737" href="Categories.Object.Product.Morphisms.html#5474" class="Function Operator">[</a> <a id="3739" href="Categories.Object.Exponential.html#3565" class="Bound">p₃</a> <a id="3742" href="Categories.Object.Product.Morphisms.html#5474" class="Function Operator">⇒</a> <a id="3744" href="Categories.Object.Exponential.html#3571" class="Bound">p₅</a> <a id="3747" href="Categories.Object.Product.Morphisms.html#5474" class="Function Operator">⇒</a> <a id="3749" href="Categories.Object.Exponential.html#4395" class="Function">p₁</a> <a id="3752" href="Categories.Object.Product.Morphisms.html#5474" class="Function Operator">]×id∘×id</a> <a id="3761" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="3765" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="3767" href="Categories.Object.Exponential.html#3559" class="Bound">e₁</a> <a id="3770" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a> <a id="3776" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3778" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="3780" href="Categories.Object.Exponential.html#3571" class="Bound">p₅</a> <a id="3783" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="3785" href="Categories.Object.Exponential.html#4395" class="Function">p₁</a> <a id="3788" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="3790" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="3792" href="Categories.Object.Exponential.html#3559" class="Bound">e₁</a> <a id="3795" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="3798" href="Categories.Object.Exponential.html#3571" class="Bound">p₅</a> <a id="3801" class="Symbol">(</a><a id="3802" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="3804" href="Categories.Object.Exponential.html#3562" class="Bound">e₂</a> <a id="3807" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a> <a id="3813" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3815" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">[</a> <a id="3817" href="Categories.Object.Exponential.html#3571" class="Bound">p₅</a> <a id="3820" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">⇒</a> <a id="3822" href="Categories.Object.Exponential.html#4426" class="Function">p₂</a> <a id="3825" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">]id×</a> <a id="3830" href="Categories.Object.Exponential.html#3575" class="Bound">f</a><a id="3831" class="Symbol">)</a> <a id="3833" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a>
<a id="3850" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3852" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="3854" href="Categories.Object.Exponential.html#3565" class="Bound">p₃</a> <a id="3857" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="3859" href="Categories.Object.Exponential.html#3571" class="Bound">p₅</a> <a id="3862" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="3864" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="3866" href="Categories.Object.Exponential.html#3562" class="Bound">e₂</a> <a id="3869" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="3872" href="Categories.Object.Exponential.html#3568" class="Bound">p₄</a> <a id="3875" href="Categories.Object.Exponential.html#3579" class="Bound">g</a> <a id="3877" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a>
<a id="3904" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3907" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="3913" class="Symbol">(</a><a id="3914" href="Categories.Object.Exponential.html#865" class="Function">e₁.β</a> <a id="3919" href="Categories.Object.Exponential.html#3571" class="Bound">p₅</a><a id="3921" class="Symbol">)</a> <a id="3923" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3927" class="Symbol">(</a><a id="3928" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="3930" href="Categories.Object.Exponential.html#3562" class="Bound">e₂</a> <a id="3933" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a> <a id="3939" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3941" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">[</a> <a id="3943" href="Categories.Object.Exponential.html#3571" class="Bound">p₅</a> <a id="3946" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">⇒</a> <a id="3948" href="Categories.Object.Exponential.html#4426" class="Function">p₂</a> <a id="3951" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">]id×</a> <a id="3956" href="Categories.Object.Exponential.html#3575" class="Bound">f</a><a id="3957" class="Symbol">)</a>
<a id="3973" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3975" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="3977" href="Categories.Object.Exponential.html#3565" class="Bound">p₃</a> <a id="3980" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="3982" href="Categories.Object.Exponential.html#3571" class="Bound">p₅</a> <a id="3985" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="3987" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="3989" href="Categories.Object.Exponential.html#3562" class="Bound">e₂</a> <a id="3992" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="3995" href="Categories.Object.Exponential.html#3568" class="Bound">p₄</a> <a id="3998" href="Categories.Object.Exponential.html#3579" class="Bound">g</a> <a id="4000" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a>
<a id="4027" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4030" href="Categories.Category.Core.html#715" class="Field">assoc</a> <a id="4036" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4040" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="4042" href="Categories.Object.Exponential.html#3562" class="Bound">e₂</a> <a id="4045" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a> <a id="4051" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4053" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">[</a> <a id="4055" href="Categories.Object.Exponential.html#3571" class="Bound">p₅</a> <a id="4058" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">⇒</a> <a id="4060" href="Categories.Object.Exponential.html#4426" class="Function">p₂</a> <a id="4063" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">]id×</a> <a id="4068" href="Categories.Object.Exponential.html#3575" class="Bound">f</a>
<a id="4083" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4085" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="4087" href="Categories.Object.Exponential.html#3565" class="Bound">p₃</a> <a id="4090" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="4092" href="Categories.Object.Exponential.html#3571" class="Bound">p₅</a> <a id="4095" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="4097" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="4099" href="Categories.Object.Exponential.html#3562" class="Bound">e₂</a> <a id="4102" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="4105" href="Categories.Object.Exponential.html#3568" class="Bound">p₄</a> <a id="4108" href="Categories.Object.Exponential.html#3579" class="Bound">g</a> <a id="4110" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a>
<a id="4137" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="4141" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="4149" href="Categories.Object.Product.Morphisms.html#6276" class="Function Operator">[</a> <a id="4151" href="Categories.Object.Exponential.html#3568" class="Bound">p₄</a> <a id="4154" href="Categories.Object.Product.Morphisms.html#6276" class="Function Operator">⇒</a> <a id="4156" href="Categories.Object.Exponential.html#4426" class="Function">p₂</a> <a id="4159" href="Categories.Object.Product.Morphisms.html#6276" class="Function Operator">,</a> <a id="4161" href="Categories.Object.Exponential.html#3565" class="Bound">p₃</a> <a id="4164" href="Categories.Object.Product.Morphisms.html#6276" class="Function Operator">⇒</a> <a id="4166" href="Categories.Object.Exponential.html#3571" class="Bound">p₅</a> <a id="4169" href="Categories.Object.Product.Morphisms.html#6276" class="Function Operator">]first↔second</a> <a id="4183" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="4187" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="4189" href="Categories.Object.Exponential.html#3562" class="Bound">e₂</a> <a id="4192" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a> <a id="4198" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4200" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="4202" href="Categories.Object.Exponential.html#3568" class="Bound">p₄</a> <a id="4205" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="4207" href="Categories.Object.Exponential.html#4426" class="Function">p₂</a> <a id="4210" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="4212" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="4214" href="Categories.Object.Exponential.html#3562" class="Bound">e₂</a> <a id="4217" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="4220" href="Categories.Object.Exponential.html#3568" class="Bound">p₄</a> <a id="4223" href="Categories.Object.Exponential.html#3579" class="Bound">g</a> <a id="4225" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a> <a id="4229" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4231" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">[</a> <a id="4233" href="Categories.Object.Exponential.html#3565" class="Bound">p₃</a> <a id="4236" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">⇒</a> <a id="4238" href="Categories.Object.Exponential.html#3568" class="Bound">p₄</a> <a id="4241" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">]id×</a> <a id="4246" href="Categories.Object.Exponential.html#3575" class="Bound">f</a>
<a id="4271" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4274" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="4280" class="Symbol">(</a><a id="4281" href="Categories.Object.Exponential.html#865" class="Function">e₂.β</a> <a id="4286" href="Categories.Object.Exponential.html#3568" class="Bound">p₄</a><a id="4288" class="Symbol">)</a> <a id="4290" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4294" href="Categories.Object.Exponential.html#3579" class="Bound">g</a> <a id="4296" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4298" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">[</a> <a id="4300" href="Categories.Object.Exponential.html#3565" class="Bound">p₃</a> <a id="4303" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">⇒</a> <a id="4305" href="Categories.Object.Exponential.html#3568" class="Bound">p₄</a> <a id="4308" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">]id×</a> <a id="4313" href="Categories.Object.Exponential.html#3575" class="Bound">f</a> <a id="4315" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="4319" class="Keyword">where</a> <a id="4325" class="Keyword">module</a> <a id="4332" href="Categories.Object.Exponential.html#4332" class="Module">e₁</a> <a id="4335" class="Symbol">=</a> <a id="4337" href="Categories.Object.Exponential.html#583" class="Module">Exponential</a> <a id="4349" href="Categories.Object.Exponential.html#3559" class="Bound">e₁</a>
<a id="4360" class="Keyword">module</a> <a id="4367" href="Categories.Object.Exponential.html#4367" class="Module">e₂</a> <a id="4370" class="Symbol">=</a> <a id="4372" href="Categories.Object.Exponential.html#583" class="Module">Exponential</a> <a id="4384" href="Categories.Object.Exponential.html#3562" class="Bound">e₂</a>
<a id="4395" href="Categories.Object.Exponential.html#4395" class="Function">p₁</a> <a id="4405" class="Symbol">=</a> <a id="4407" href="Categories.Object.Exponential.html#657" class="Function">e₁.product</a>
<a id="4426" href="Categories.Object.Exponential.html#4426" class="Function">p₂</a> <a id="4436" class="Symbol">=</a> <a id="4438" href="Categories.Object.Exponential.html#657" class="Function">e₂.product</a>
<a id="repack"></a><a id="4450" href="Categories.Object.Exponential.html#4450" class="Function">repack</a> <a id="4457" class="Symbol">:</a> <a id="4459" class="Symbol">∀{</a><a id="4461" href="Categories.Object.Exponential.html#4461" class="Bound">A</a> <a id="4463" href="Categories.Object.Exponential.html#4463" class="Bound">B</a><a id="4464" class="Symbol">}</a> <a id="4466" class="Symbol">(</a><a id="4467" href="Categories.Object.Exponential.html#4467" class="Bound">e₁</a> <a id="4470" href="Categories.Object.Exponential.html#4470" class="Bound">e₂</a> <a id="4473" class="Symbol">:</a> <a id="4475" href="Categories.Object.Exponential.html#583" class="Record">Exponential</a> <a id="4487" href="Categories.Object.Exponential.html#4461" class="Bound">A</a> <a id="4489" href="Categories.Object.Exponential.html#4463" class="Bound">B</a><a id="4490" class="Symbol">)</a> <a id="4492" class="Symbol">→</a> <a id="4494" href="Categories.Object.Exponential.html#643" class="Field">Exponential.B^A</a> <a id="4510" href="Categories.Object.Exponential.html#4467" class="Bound">e₁</a> <a id="4513" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="4515" href="Categories.Object.Exponential.html#643" class="Field">Exponential.B^A</a> <a id="4531" href="Categories.Object.Exponential.html#4470" class="Bound">e₂</a>
<a id="4534" href="Categories.Object.Exponential.html#4450" class="Function">repack</a> <a id="4541" href="Categories.Object.Exponential.html#4541" class="Bound">e₁</a> <a id="4544" href="Categories.Object.Exponential.html#4544" class="Bound">e₂</a> <a id="4547" class="Symbol">=</a> <a id="4549" href="Categories.Object.Exponential.html#792" class="Field">e₂.λg</a> <a id="4555" href="Categories.Object.Exponential.html#657" class="Function">e₁.product</a> <a id="4566" href="Categories.Object.Exponential.html#767" class="Function">e₁.eval</a>
<a id="4576" class="Keyword">where</a>
<a id="4586" class="Keyword">module</a> <a id="4593" href="Categories.Object.Exponential.html#4593" class="Module">e₁</a> <a id="4596" class="Symbol">=</a> <a id="4598" href="Categories.Object.Exponential.html#583" class="Module">Exponential</a> <a id="4610" href="Categories.Object.Exponential.html#4541" class="Bound">e₁</a>
<a id="4617" class="Keyword">module</a> <a id="4624" href="Categories.Object.Exponential.html#4624" class="Module">e₂</a> <a id="4627" class="Symbol">=</a> <a id="4629" href="Categories.Object.Exponential.html#583" class="Module">Exponential</a> <a id="4641" href="Categories.Object.Exponential.html#4544" class="Bound">e₂</a>
<a id="repack≡id"></a><a id="4645" href="Categories.Object.Exponential.html#4645" class="Function">repack≡id</a> <a id="4655" class="Symbol">:</a> <a id="4657" class="Symbol">∀{</a><a id="4659" href="Categories.Object.Exponential.html#4659" class="Bound">A</a> <a id="4661" href="Categories.Object.Exponential.html#4661" class="Bound">B</a><a id="4662" class="Symbol">}</a> <a id="4664" class="Symbol">(</a><a id="4665" href="Categories.Object.Exponential.html#4665" class="Bound">e</a> <a id="4667" class="Symbol">:</a> <a id="4669" href="Categories.Object.Exponential.html#583" class="Record">Exponential</a> <a id="4681" href="Categories.Object.Exponential.html#4659" class="Bound">A</a> <a id="4683" href="Categories.Object.Exponential.html#4661" class="Bound">B</a><a id="4684" class="Symbol">)</a> <a id="4686" class="Symbol">→</a> <a id="4688" href="Categories.Object.Exponential.html#4450" class="Function">repack</a> <a id="4695" href="Categories.Object.Exponential.html#4665" class="Bound">e</a> <a id="4697" href="Categories.Object.Exponential.html#4665" class="Bound">e</a> <a id="4699" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="4701" href="Categories.Category.Core.html#630" class="Field">id</a>
<a id="4704" href="Categories.Object.Exponential.html#4645" class="Function">repack≡id</a> <a id="4714" class="Symbol">=</a> <a id="4716" href="Categories.Object.Exponential.html#2201" class="Function">Exponential.η-id</a>
<a id="repack∘"></a><a id="4734" href="Categories.Object.Exponential.html#4734" class="Function">repack∘</a> <a id="4742" class="Symbol">:</a> <a id="4744" class="Symbol">∀</a> <a id="4746" class="Symbol">(</a><a id="4747" href="Categories.Object.Exponential.html#4747" class="Bound">e₁</a> <a id="4750" href="Categories.Object.Exponential.html#4750" class="Bound">e₂</a> <a id="4753" href="Categories.Object.Exponential.html#4753" class="Bound">e₃</a> <a id="4756" class="Symbol">:</a> <a id="4758" href="Categories.Object.Exponential.html#583" class="Record">Exponential</a> <a id="4770" href="Categories.Object.Exponential.html#557" class="Generalizable">A</a> <a id="4772" href="Categories.Object.Exponential.html#559" class="Generalizable">B</a><a id="4773" class="Symbol">)</a> <a id="4775" class="Symbol">→</a> <a id="4777" href="Categories.Object.Exponential.html#4450" class="Function">repack</a> <a id="4784" href="Categories.Object.Exponential.html#4750" class="Bound">e₂</a> <a id="4787" href="Categories.Object.Exponential.html#4753" class="Bound">e₃</a> <a id="4790" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4792" href="Categories.Object.Exponential.html#4450" class="Function">repack</a> <a id="4799" href="Categories.Object.Exponential.html#4747" class="Bound">e₁</a> <a id="4802" href="Categories.Object.Exponential.html#4750" class="Bound">e₂</a> <a id="4805" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="4807" href="Categories.Object.Exponential.html#4450" class="Function">repack</a> <a id="4814" href="Categories.Object.Exponential.html#4747" class="Bound">e₁</a> <a id="4817" href="Categories.Object.Exponential.html#4753" class="Bound">e₃</a>
<a id="4820" href="Categories.Object.Exponential.html#4734" class="Function">repack∘</a> <a id="4828" href="Categories.Object.Exponential.html#4828" class="Bound">e₁</a> <a id="4831" href="Categories.Object.Exponential.html#4831" class="Bound">e₂</a> <a id="4834" href="Categories.Object.Exponential.html#4834" class="Bound">e₃</a> <a id="4837" class="Symbol">=</a>
<a id="4841" class="Keyword">let</a> <a id="4845" href="Categories.Object.Exponential.html#4845" class="Bound">p₁</a> <a id="4848" class="Symbol">=</a> <a id="4850" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="4858" href="Categories.Object.Exponential.html#4828" class="Bound">e₁</a> <a id="4861" class="Keyword">in</a>
<a id="4866" class="Keyword">let</a> <a id="4870" href="Categories.Object.Exponential.html#4870" class="Bound">p₂</a> <a id="4873" class="Symbol">=</a> <a id="4875" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="4883" href="Categories.Object.Exponential.html#4831" class="Bound">e₂</a> <a id="4886" class="Keyword">in</a>
<a id="4891" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="4903" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="4905" href="Categories.Object.Exponential.html#4834" class="Bound">e₃</a> <a id="4908" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="4911" href="Categories.Object.Exponential.html#4870" class="Bound">p₂</a> <a id="4914" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="4916" href="Categories.Object.Exponential.html#4831" class="Bound">e₂</a> <a id="4919" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a>
<a id="4929" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4931" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="4933" href="Categories.Object.Exponential.html#4831" class="Bound">e₂</a> <a id="4936" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="4939" href="Categories.Object.Exponential.html#4845" class="Bound">p₁</a> <a id="4942" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="4944" href="Categories.Object.Exponential.html#4828" class="Bound">e₁</a> <a id="4947" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a>
<a id="4955" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4958" href="Categories.Object.Exponential.html#1272" class="Function">λ-cong</a> <a id="4965" href="Categories.Object.Exponential.html#4834" class="Bound">e₃</a> <a id="4968" href="Categories.Object.Exponential.html#4870" class="Bound">p₂</a> <a id="4971" class="Symbol">(</a><a id="4972" href="Categories.Morphism.Reasoning.Core.html#2899" class="Function">introʳ</a> <a id="4979" class="Symbol">(</a><a id="4980" href="Categories.Object.Product.Morphisms.html#4591" class="Function">second-id</a> <a id="4990" href="Categories.Object.Exponential.html#4870" class="Bound">p₂</a><a id="4992" class="Symbol">))</a> <a id="4995" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="5003" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5011" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="5013" href="Categories.Object.Exponential.html#4834" class="Bound">e₃</a> <a id="5016" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="5019" href="Categories.Object.Exponential.html#4870" class="Bound">p₂</a> <a id="5022" class="Symbol">(</a><a id="5023" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="5025" href="Categories.Object.Exponential.html#4831" class="Bound">e₂</a> <a id="5028" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a> <a id="5034" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5036" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">[</a> <a id="5038" href="Categories.Object.Exponential.html#4870" class="Bound">p₂</a> <a id="5041" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">⇒</a> <a id="5043" href="Categories.Object.Exponential.html#4870" class="Bound">p₂</a> <a id="5046" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">]id×</a> <a id="5051" href="Categories.Category.Core.html#630" class="Field">id</a><a id="5053" class="Symbol">)</a>
<a id="5059" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5061" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="5063" href="Categories.Object.Exponential.html#4831" class="Bound">e₂</a> <a id="5066" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="5069" href="Categories.Object.Exponential.html#4845" class="Bound">p₁</a> <a id="5072" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="5074" href="Categories.Object.Exponential.html#4828" class="Bound">e₁</a> <a id="5077" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a>
<a id="5085" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="5089" href="Categories.Object.Exponential.html#3212" class="Function">λ-distrib</a> <a id="5099" href="Categories.Object.Exponential.html#4834" class="Bound">e₃</a> <a id="5102" href="Categories.Object.Exponential.html#4831" class="Bound">e₂</a> <a id="5105" href="Categories.Object.Exponential.html#4845" class="Bound">p₁</a> <a id="5108" href="Categories.Object.Exponential.html#4845" class="Bound">p₁</a> <a id="5111" href="Categories.Object.Exponential.html#4870" class="Bound">p₂</a> <a id="5114" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="5120" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="5122" href="Categories.Object.Exponential.html#4834" class="Bound">e₃</a> <a id="5125" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="5128" href="Categories.Object.Exponential.html#4845" class="Bound">p₁</a> <a id="5131" class="Symbol">(</a><a id="5132" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="5134" href="Categories.Object.Exponential.html#4828" class="Bound">e₁</a> <a id="5137" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a> <a id="5143" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5145" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">[</a> <a id="5147" href="Categories.Object.Exponential.html#4845" class="Bound">p₁</a> <a id="5150" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">⇒</a> <a id="5152" href="Categories.Object.Exponential.html#4845" class="Bound">p₁</a> <a id="5155" href="Categories.Object.Product.Morphisms.html#4398" class="Function Operator">]id×</a> <a id="5160" href="Categories.Category.Core.html#630" class="Field">id</a><a id="5162" class="Symbol">)</a>
<a id="5166" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5169" href="Categories.Object.Exponential.html#1272" class="Function">λ-cong</a> <a id="5176" href="Categories.Object.Exponential.html#4834" class="Bound">e₃</a> <a id="5179" href="Categories.Object.Exponential.html#4845" class="Bound">p₁</a> <a id="5182" class="Symbol">(</a><a id="5183" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="5185" class="Symbol">(</a><a id="5186" href="Categories.Morphism.Reasoning.Core.html#2899" class="Function">introʳ</a> <a id="5193" class="Symbol">(</a><a id="5194" href="Categories.Object.Product.Morphisms.html#4591" class="Function">second-id</a> <a id="5204" class="Symbol">(</a><a id="5205" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="5213" href="Categories.Object.Exponential.html#4828" class="Bound">e₁</a><a id="5215" class="Symbol">))))</a> <a id="5220" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5226" href="Categories.Object.Exponential.html#2862" class="Function Operator">[</a> <a id="5228" href="Categories.Object.Exponential.html#4834" class="Bound">e₃</a> <a id="5231" href="Categories.Object.Exponential.html#2862" class="Function Operator">]λ</a> <a id="5234" href="Categories.Object.Exponential.html#4845" class="Bound">p₁</a> <a id="5237" href="Categories.Object.Exponential.html#2762" class="Function Operator">[</a> <a id="5239" href="Categories.Object.Exponential.html#4828" class="Bound">e₁</a> <a id="5242" href="Categories.Object.Exponential.html#2762" class="Function Operator">]eval</a>
<a id="5250" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="5254" class="Keyword">where</a> <a id="5260" class="Keyword">open</a> <a id="5265" href="Categories.Object.Exponential.html#583" class="Module">Exponential</a>
<a id="repack-cancel"></a><a id="5278" href="Categories.Object.Exponential.html#5278" class="Function">repack-cancel</a> <a id="5292" class="Symbol">:</a> <a id="5294" class="Symbol">∀{</a><a id="5296" href="Categories.Object.Exponential.html#5296" class="Bound">A</a> <a id="5298" href="Categories.Object.Exponential.html#5298" class="Bound">B</a><a id="5299" class="Symbol">}</a> <a id="5301" class="Symbol">(</a><a id="5302" href="Categories.Object.Exponential.html#5302" class="Bound">e₁</a> <a id="5305" href="Categories.Object.Exponential.html#5305" class="Bound">e₂</a> <a id="5308" class="Symbol">:</a> <a id="5310" href="Categories.Object.Exponential.html#583" class="Record">Exponential</a> <a id="5322" href="Categories.Object.Exponential.html#5296" class="Bound">A</a> <a id="5324" href="Categories.Object.Exponential.html#5298" class="Bound">B</a><a id="5325" class="Symbol">)</a> <a id="5327" class="Symbol">→</a> <a id="5329" href="Categories.Object.Exponential.html#4450" class="Function">repack</a> <a id="5336" href="Categories.Object.Exponential.html#5302" class="Bound">e₁</a> <a id="5339" href="Categories.Object.Exponential.html#5305" class="Bound">e₂</a> <a id="5342" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5344" href="Categories.Object.Exponential.html#4450" class="Function">repack</a> <a id="5351" href="Categories.Object.Exponential.html#5305" class="Bound">e₂</a> <a id="5354" href="Categories.Object.Exponential.html#5302" class="Bound">e₁</a> <a id="5357" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="5359" href="Categories.Category.Core.html#630" class="Field">id</a>
<a id="5362" href="Categories.Object.Exponential.html#5278" class="Function">repack-cancel</a> <a id="5376" href="Categories.Object.Exponential.html#5376" class="Bound">e₁</a> <a id="5379" href="Categories.Object.Exponential.html#5379" class="Bound">e₂</a> <a id="5382" class="Symbol">=</a> <a id="5384" href="Categories.Object.Exponential.html#4734" class="Function">repack∘</a> <a id="5392" href="Categories.Object.Exponential.html#5379" class="Bound">e₂</a> <a id="5395" href="Categories.Object.Exponential.html#5376" class="Bound">e₁</a> <a id="5398" href="Categories.Object.Exponential.html#5379" class="Bound">e₂</a> <a id="5401" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="5403" href="Categories.Object.Exponential.html#4645" class="Function">repack≡id</a> <a id="5413" href="Categories.Object.Exponential.html#5379" class="Bound">e₂</a>
<a id="up-to-iso"></a><a id="5417" href="Categories.Object.Exponential.html#5417" class="Function">up-to-iso</a> <a id="5427" class="Symbol">:</a> <a id="5429" class="Symbol">∀{</a><a id="5431" href="Categories.Object.Exponential.html#5431" class="Bound">A</a> <a id="5433" href="Categories.Object.Exponential.html#5433" class="Bound">B</a><a id="5434" class="Symbol">}</a> <a id="5436" class="Symbol">(</a><a id="5437" href="Categories.Object.Exponential.html#5437" class="Bound">e₁</a> <a id="5440" href="Categories.Object.Exponential.html#5440" class="Bound">e₂</a> <a id="5443" class="Symbol">:</a> <a id="5445" href="Categories.Object.Exponential.html#583" class="Record">Exponential</a> <a id="5457" href="Categories.Object.Exponential.html#5431" class="Bound">A</a> <a id="5459" href="Categories.Object.Exponential.html#5433" class="Bound">B</a><a id="5460" class="Symbol">)</a> <a id="5462" class="Symbol">→</a> <a id="5464" href="Categories.Object.Exponential.html#643" class="Field">Exponential.B^A</a> <a id="5480" href="Categories.Object.Exponential.html#5437" class="Bound">e₁</a> <a id="5483" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="5485" href="Categories.Object.Exponential.html#643" class="Field">Exponential.B^A</a> <a id="5501" href="Categories.Object.Exponential.html#5440" class="Bound">e₂</a>
<a id="5504" href="Categories.Object.Exponential.html#5417" class="Function">up-to-iso</a> <a id="5514" href="Categories.Object.Exponential.html#5514" class="Bound">e₁</a> <a id="5517" href="Categories.Object.Exponential.html#5517" class="Bound">e₂</a> <a id="5520" class="Symbol">=</a> <a id="5522" class="Keyword">record</a>
<a id="5531" class="Symbol">{</a> <a id="5533" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="5538" class="Symbol">=</a> <a id="5540" href="Categories.Object.Exponential.html#4450" class="Function">repack</a> <a id="5547" href="Categories.Object.Exponential.html#5514" class="Bound">e₁</a> <a id="5550" href="Categories.Object.Exponential.html#5517" class="Bound">e₂</a>
<a id="5555" class="Symbol">;</a> <a id="5557" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="5562" class="Symbol">=</a> <a id="5564" href="Categories.Object.Exponential.html#4450" class="Function">repack</a> <a id="5571" href="Categories.Object.Exponential.html#5517" class="Bound">e₂</a> <a id="5574" href="Categories.Object.Exponential.html#5514" class="Bound">e₁</a>
<a id="5579" class="Symbol">;</a> <a id="5581" href="Categories.Morphism.html#2040" class="Field">iso</a> <a id="5586" class="Symbol">=</a> <a id="5588" class="Keyword">record</a>
<a id="5599" class="Symbol">{</a> <a id="5601" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="5606" class="Symbol">=</a> <a id="5608" href="Categories.Object.Exponential.html#5278" class="Function">repack-cancel</a> <a id="5622" href="Categories.Object.Exponential.html#5517" class="Bound">e₂</a> <a id="5625" href="Categories.Object.Exponential.html#5514" class="Bound">e₁</a>
<a id="5632" class="Symbol">;</a> <a id="5634" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="5639" class="Symbol">=</a> <a id="5641" href="Categories.Object.Exponential.html#5278" class="Function">repack-cancel</a> <a id="5655" href="Categories.Object.Exponential.html#5514" class="Bound">e₁</a> <a id="5658" href="Categories.Object.Exponential.html#5517" class="Bound">e₂</a>
<a id="5665" class="Symbol">}</a>
<a id="5669" class="Symbol">}</a>
<a id="transport-by-iso"></a><a id="5672" href="Categories.Object.Exponential.html#5672" class="Function">transport-by-iso</a> <a id="5689" class="Symbol">:</a> <a id="5691" class="Symbol">∀</a> <a id="5693" class="Symbol">(</a><a id="5694" href="Categories.Object.Exponential.html#5694" class="Bound">e</a> <a id="5696" class="Symbol">:</a> <a id="5698" href="Categories.Object.Exponential.html#583" class="Record">Exponential</a> <a id="5710" href="Categories.Object.Exponential.html#557" class="Generalizable">A</a> <a id="5712" href="Categories.Object.Exponential.html#559" class="Generalizable">B</a><a id="5713" class="Symbol">)</a> <a id="5715" class="Symbol">→</a> <a id="5717" href="Categories.Object.Exponential.html#643" class="Field">Exponential.B^A</a> <a id="5733" href="Categories.Object.Exponential.html#5694" class="Bound">e</a> <a id="5735" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="5737" href="Categories.Object.Exponential.html#565" class="Generalizable">X</a> <a id="5739" class="Symbol">→</a> <a id="5741" href="Categories.Object.Exponential.html#583" class="Record">Exponential</a> <a id="5753" href="Categories.Object.Exponential.html#557" class="Generalizable">A</a> <a id="5755" href="Categories.Object.Exponential.html#559" class="Generalizable">B</a>
<a id="5757" href="Categories.Object.Exponential.html#5672" class="Function">transport-by-iso</a> <a id="5774" class="Symbol">{</a><a id="5775" class="Argument">X</a> <a id="5777" class="Symbol">=</a> <a id="5779" href="Categories.Object.Exponential.html#5779" class="Bound">X</a><a id="5780" class="Symbol">}</a> <a id="5782" href="Categories.Object.Exponential.html#5782" class="Bound">e</a> <a id="5784" href="Categories.Object.Exponential.html#5784" class="Bound">e≅X</a> <a id="5788" class="Symbol">=</a> <a id="5790" class="Keyword">record</a>
<a id="5799" class="Symbol">{</a> <a id="5801" href="Categories.Object.Exponential.html#643" class="Field">B^A</a> <a id="5817" class="Symbol">=</a> <a id="5819" href="Categories.Object.Exponential.html#5779" class="Bound">X</a>
<a id="5823" class="Symbol">;</a> <a id="5825" href="Categories.Object.Exponential.html#657" class="Field">product</a> <a id="5841" class="Symbol">=</a> <a id="5843" href="Categories.Object.Exponential.html#6639" class="Function">X×A</a>
<a id="5849" class="Symbol">;</a> <a id="5851" href="Categories.Object.Exponential.html#767" class="Field">eval</a> <a id="5867" class="Symbol">=</a> <a id="5869" href="Categories.Object.Exponential.html#767" class="Function">e.eval</a>
<a id="5878" class="Symbol">;</a> <a id="5880" href="Categories.Object.Exponential.html#792" class="Field">λg</a> <a id="5896" class="Symbol">=</a> <a id="5898" class="Symbol">λ</a> <a id="5900" href="Categories.Object.Exponential.html#5900" class="Bound">Y×A</a> <a id="5904" href="Categories.Object.Exponential.html#5904" class="Bound">Y×A⇒B</a> <a id="5910" class="Symbol">→</a> <a id="5912" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="5917" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5919" class="Symbol">(</a><a id="5920" href="Categories.Object.Exponential.html#792" class="Function">e.λg</a> <a id="5925" href="Categories.Object.Exponential.html#5900" class="Bound">Y×A</a> <a id="5929" href="Categories.Object.Exponential.html#5904" class="Bound">Y×A⇒B</a><a id="5934" class="Symbol">)</a>
<a id="5938" class="Symbol">;</a> <a id="5940" href="Categories.Object.Exponential.html#865" class="Field">β</a> <a id="5956" class="Symbol">=</a> <a id="5958" class="Symbol">λ</a> <a id="5960" href="Categories.Object.Exponential.html#5960" class="Bound">Y×A</a> <a id="5964" class="Symbol">{</a><a id="5965" href="Categories.Object.Exponential.html#5965" class="Bound">h</a><a id="5966" class="Symbol">}</a> <a id="5968" class="Symbol">→</a> <a id="5970" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="5980" href="Categories.Object.Exponential.html#767" class="Function">e.eval</a> <a id="5987" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5989" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="5991" href="Categories.Object.Exponential.html#5960" class="Bound">Y×A</a> <a id="5995" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="5997" href="Categories.Object.Exponential.html#6639" class="Function">X×A</a> <a id="6001" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="6003" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="6008" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="6010" href="Categories.Object.Exponential.html#792" class="Function">e.λg</a> <a id="6015" href="Categories.Object.Exponential.html#5960" class="Bound">Y×A</a> <a id="6019" href="Categories.Object.Exponential.html#5965" class="Bound">h</a> <a id="6021" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a> <a id="6025" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6028" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="6036" href="Categories.Object.Product.Core.html#854" class="Function">e.product.⟨⟩-cong₂</a> <a id="6055" class="Symbol">(</a><a id="6056" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="6062" class="Symbol">(</a><a id="6063" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="6071" href="Categories.Morphism.html#1586" class="Function">isoˡ</a><a id="6075" class="Symbol">))</a>
<a id="6156" class="Symbol">(</a><a id="6157" href="Categories.Morphism.Reasoning.Core.html#2948" class="Function">elimˡ</a> <a id="6163" href="Relation.Binary.Structures.html#1596" class="Function">Equiv.refl</a><a id="6173" class="Symbol">)</a> <a id="6175" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6181" href="Categories.Object.Exponential.html#767" class="Function">e.eval</a> <a id="6188" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="6190" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="6192" href="Categories.Object.Exponential.html#5960" class="Bound">Y×A</a> <a id="6196" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="6198" href="Categories.Object.Exponential.html#657" class="Function">e.product</a> <a id="6208" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="6210" href="Categories.Object.Exponential.html#792" class="Function">e.λg</a> <a id="6215" href="Categories.Object.Exponential.html#5960" class="Bound">Y×A</a> <a id="6219" href="Categories.Object.Exponential.html#5965" class="Bound">h</a> <a id="6221" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a> <a id="6226" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6229" href="Categories.Object.Exponential.html#865" class="Function">e.β</a> <a id="6233" href="Categories.Object.Exponential.html#5960" class="Bound">Y×A</a> <a id="6237" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6243" href="Categories.Object.Exponential.html#5965" class="Bound">h</a> <a id="6288" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="6292" class="Symbol">;</a> <a id="6294" href="Categories.Object.Exponential.html#991" class="Field">λ-unique</a> <a id="6310" class="Symbol">=</a> <a id="6312" class="Symbol">λ</a> <a id="6314" href="Categories.Object.Exponential.html#6314" class="Bound">Y×A</a> <a id="6318" class="Symbol">{</a><a id="6319" href="Categories.Object.Exponential.html#6319" class="Bound">h</a><a id="6320" class="Symbol">}</a> <a id="6322" class="Symbol">{</a><a id="6323" href="Categories.Object.Exponential.html#6323" class="Bound">i</a><a id="6324" class="Symbol">}</a> <a id="6326" href="Categories.Object.Exponential.html#6326" class="Bound">eq</a> <a id="6329" class="Symbol">→</a>
<a id="6335" href="Categories.Morphism.Reasoning.Iso.html#907" class="Function">switch-tofromˡ</a> <a id="6350" href="Categories.Object.Exponential.html#5784" class="Bound">e≅X</a> <a id="6354" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="6356" href="Categories.Object.Exponential.html#991" class="Function">e.λ-unique</a> <a id="6367" href="Categories.Object.Exponential.html#6314" class="Bound">Y×A</a> <a id="6371" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="6373" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="6385" href="Categories.Object.Exponential.html#767" class="Function">e.eval</a> <a id="6392" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="6394" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="6396" href="Categories.Object.Exponential.html#6314" class="Bound">Y×A</a> <a id="6400" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="6402" href="Categories.Object.Exponential.html#657" class="Function">e.product</a> <a id="6412" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="6414" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="6417" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="6419" href="Categories.Object.Exponential.html#6323" class="Bound">i</a> <a id="6421" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a> <a id="6428" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6431" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="6439" href="Categories.Object.Product.Core.html#854" class="Function">e.product.⟨⟩-cong₂</a> <a id="6458" href="Categories.Category.Core.html#715" class="Field">assoc</a> <a id="6464" class="Symbol">(</a><a id="6465" href="Categories.Morphism.Reasoning.Core.html#3063" class="Function">introˡ</a> <a id="6472" href="Relation.Binary.Structures.html#1596" class="Function">Equiv.refl</a><a id="6482" class="Symbol">)</a> <a id="6484" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6492" href="Categories.Object.Exponential.html#767" class="Function">e.eval</a> <a id="6499" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="6501" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">[</a> <a id="6503" href="Categories.Object.Exponential.html#6314" class="Bound">Y×A</a> <a id="6507" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">⇒</a> <a id="6509" href="Categories.Object.Exponential.html#6639" class="Function">X×A</a> <a id="6513" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">]</a> <a id="6515" href="Categories.Object.Exponential.html#6323" class="Bound">i</a> <a id="6517" href="Categories.Object.Product.Morphisms.html#4276" class="Function Operator">×id</a> <a id="6535" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6538" href="Categories.Object.Exponential.html#6326" class="Bound">eq</a> <a id="6541" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6549" href="Categories.Object.Exponential.html#6319" class="Bound">h</a> <a id="6592" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="6596" class="Symbol">}</a>
<a id="6600" class="Keyword">where</a> <a id="6606" class="Keyword">module</a> <a id="6613" href="Categories.Object.Exponential.html#6613" class="Module">e</a> <a id="6615" class="Symbol">=</a> <a id="6617" href="Categories.Object.Exponential.html#583" class="Module">Exponential</a> <a id="6629" href="Categories.Object.Exponential.html#5782" class="Bound">e</a>
<a id="6639" href="Categories.Object.Exponential.html#6639" class="Function">X×A</a> <a id="6648" class="Symbol">=</a> <a id="6650" href="Categories.Object.Product.Core.html#5921" class="Function">Mobile</a> <a id="6657" href="Categories.Object.Exponential.html#657" class="Function">e.product</a> <a id="6667" href="Categories.Object.Exponential.html#5784" class="Bound">e≅X</a> <a id="6671" href="Relation.Binary.Structures.html#1596" class="Function">≅.refl</a>
<a id="6686" class="Keyword">open</a> <a id="6691" href="Categories.Morphism.html#1958" class="Module Operator">_≅_</a> <a id="6695" href="Categories.Object.Exponential.html#5784" class="Bound">e≅X</a>
</pre></body></html>
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