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<html><head><meta charset="utf-8"><title>Categories.Adjoint.TwoSided.Compose</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="37" class="Keyword">module</a> <a id="44" href="Categories.Adjoint.TwoSided.Compose.html" class="Module">Categories.Adjoint.TwoSided.Compose</a> <a id="80" class="Keyword">where</a>
<a id="87" class="Keyword">open</a> <a id="92" class="Keyword">import</a> <a id="99" href="Level.html" class="Module">Level</a>
<a id="106" class="Keyword">open</a> <a id="111" class="Keyword">import</a> <a id="118" href="Categories.Adjoint.html" class="Module">Categories.Adjoint</a>
<a id="137" class="Keyword">open</a> <a id="142" class="Keyword">import</a> <a id="149" href="Categories.Adjoint.TwoSided.html" class="Module">Categories.Adjoint.TwoSided</a>
<a id="177" class="Keyword">open</a> <a id="182" class="Keyword">import</a> <a id="189" href="Categories.Category.Core.html" class="Module">Categories.Category.Core</a> <a id="214" class="Keyword">using</a> <a id="220" class="Symbol">(</a><a id="221" href="Categories.Category.Core.html#442" class="Record">Category</a><a id="229" class="Symbol">)</a>
<a id="231" class="Keyword">open</a> <a id="236" class="Keyword">import</a> <a id="243" href="Categories.Functor.html" class="Module">Categories.Functor</a> <a id="262" class="Keyword">renaming</a> <a id="271" class="Symbol">(</a><a id="272" href="Categories.Functor.html#349" class="Function">id</a> <a id="275" class="Symbol">to</a> <a id="278" class="Function">idF</a><a id="281" class="Symbol">)</a>
<a id="283" class="Keyword">open</a> <a id="288" class="Keyword">import</a> <a id="295" href="Categories.Functor.Properties.html" class="Module">Categories.Functor.Properties</a>
<a id="325" class="Keyword">open</a> <a id="330" class="Keyword">import</a> <a id="337" href="Categories.NaturalTransformation.html" class="Module">Categories.NaturalTransformation</a> <a id="370" class="Keyword">using</a> <a id="376" class="Symbol">(</a><a id="377" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a><a id="385" class="Symbol">)</a>
<a id="387" class="Keyword">open</a> <a id="392" class="Keyword">import</a> <a id="399" href="Categories.NaturalTransformation.NaturalIsomorphism.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism</a> <a id="451" class="Symbol">as</a> <a id="454" class="Module">≃</a> <a id="456" class="Keyword">using</a> <a id="462" class="Symbol">(</a><a id="463" href="Categories.NaturalTransformation.NaturalIsomorphism.html#3600" class="Function Operator">_≃_</a><a id="466" class="Symbol">;</a> <a id="468" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a><a id="486" class="Symbol">)</a>
<a id="488" class="Keyword">open</a> <a id="493" class="Keyword">import</a> <a id="500" href="Categories.NaturalTransformation.NaturalIsomorphism.Properties.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism.Properties</a>
<a id="563" class="Keyword">import</a> <a id="570" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="600" class="Symbol">as</a> <a id="603" class="Module">MR</a>
<a id="607" class="Keyword">open</a> <a id="612" class="Keyword">import</a> <a id="619" href="Relation.Binary.html" class="Module">Relation.Binary</a> <a id="635" class="Keyword">using</a> <a id="641" class="Symbol">(</a><a id="642" href="Relation.Binary.Bundles.html#1018" class="Record">Setoid</a><a id="648" class="Symbol">;</a> <a id="650" href="Relation.Binary.Structures.html#1531" class="Record">IsEquivalence</a><a id="663" class="Symbol">)</a>
<a id="666" class="Keyword">private</a>
<a id="676" class="Keyword">variable</a>
<a id="689" href="Categories.Adjoint.TwoSided.Compose.html#689" class="Generalizable">o</a> <a id="691" href="Categories.Adjoint.TwoSided.Compose.html#691" class="Generalizable">ℓ</a> <a id="693" href="Categories.Adjoint.TwoSided.Compose.html#693" class="Generalizable">e</a> <a id="695" href="Categories.Adjoint.TwoSided.Compose.html#695" class="Generalizable">o′</a> <a id="698" href="Categories.Adjoint.TwoSided.Compose.html#698" class="Generalizable">ℓ′</a> <a id="701" href="Categories.Adjoint.TwoSided.Compose.html#701" class="Generalizable">e′</a> <a id="704" href="Categories.Adjoint.TwoSided.Compose.html#704" class="Generalizable">o″</a> <a id="707" href="Categories.Adjoint.TwoSided.Compose.html#707" class="Generalizable">ℓ″</a> <a id="710" href="Categories.Adjoint.TwoSided.Compose.html#710" class="Generalizable">e″</a> <a id="713" class="Symbol">:</a> <a id="715" href="Agda.Primitive.html#591" class="Postulate">Level</a>
<a id="_∘⊣⊢_"></a><a id="722" href="Categories.Adjoint.TwoSided.Compose.html#722" class="Function Operator">_∘⊣⊢_</a> <a id="728" class="Symbol">:</a> <a id="730" class="Symbol">{</a><a id="731" href="Categories.Adjoint.TwoSided.Compose.html#731" class="Bound">C</a> <a id="733" class="Symbol">:</a> <a id="735" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="744" href="Categories.Adjoint.TwoSided.Compose.html#689" class="Generalizable">o</a> <a id="746" href="Categories.Adjoint.TwoSided.Compose.html#691" class="Generalizable">ℓ</a> <a id="748" href="Categories.Adjoint.TwoSided.Compose.html#693" class="Generalizable">e</a><a id="749" class="Symbol">}</a> <a id="751" class="Symbol">{</a><a id="752" href="Categories.Adjoint.TwoSided.Compose.html#752" class="Bound">D</a> <a id="754" class="Symbol">:</a> <a id="756" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="765" href="Categories.Adjoint.TwoSided.Compose.html#695" class="Generalizable">o′</a> <a id="768" href="Categories.Adjoint.TwoSided.Compose.html#698" class="Generalizable">ℓ′</a> <a id="771" href="Categories.Adjoint.TwoSided.Compose.html#701" class="Generalizable">e′</a><a id="773" class="Symbol">}</a> <a id="775" class="Symbol">{</a><a id="776" href="Categories.Adjoint.TwoSided.Compose.html#776" class="Bound">E</a> <a id="778" class="Symbol">:</a> <a id="780" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="789" href="Categories.Adjoint.TwoSided.Compose.html#704" class="Generalizable">o″</a> <a id="792" href="Categories.Adjoint.TwoSided.Compose.html#707" class="Generalizable">ℓ″</a> <a id="795" href="Categories.Adjoint.TwoSided.Compose.html#710" class="Generalizable">e″</a><a id="797" class="Symbol">}</a>
<a id="807" class="Symbol">{</a><a id="808" href="Categories.Adjoint.TwoSided.Compose.html#808" class="Bound">L</a> <a id="810" class="Symbol">:</a> <a id="812" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="820" href="Categories.Adjoint.TwoSided.Compose.html#731" class="Bound">C</a> <a id="822" href="Categories.Adjoint.TwoSided.Compose.html#752" class="Bound">D</a><a id="823" class="Symbol">}</a> <a id="825" class="Symbol">{</a><a id="826" href="Categories.Adjoint.TwoSided.Compose.html#826" class="Bound">R</a> <a id="828" class="Symbol">:</a> <a id="830" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="838" href="Categories.Adjoint.TwoSided.Compose.html#752" class="Bound">D</a> <a id="840" href="Categories.Adjoint.TwoSided.Compose.html#731" class="Bound">C</a><a id="841" class="Symbol">}</a> <a id="843" class="Symbol">{</a><a id="844" href="Categories.Adjoint.TwoSided.Compose.html#844" class="Bound">L′</a> <a id="847" class="Symbol">:</a> <a id="849" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="857" href="Categories.Adjoint.TwoSided.Compose.html#752" class="Bound">D</a> <a id="859" href="Categories.Adjoint.TwoSided.Compose.html#776" class="Bound">E</a><a id="860" class="Symbol">}</a> <a id="862" class="Symbol">{</a><a id="863" href="Categories.Adjoint.TwoSided.Compose.html#863" class="Bound">R′</a> <a id="866" class="Symbol">:</a> <a id="868" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="876" href="Categories.Adjoint.TwoSided.Compose.html#776" class="Bound">E</a> <a id="878" href="Categories.Adjoint.TwoSided.Compose.html#752" class="Bound">D</a><a id="879" class="Symbol">}</a>
<a id="889" class="Symbol">(</a><a id="890" href="Categories.Adjoint.TwoSided.Compose.html#890" class="Bound">L⊣⊢R</a> <a id="895" class="Symbol">:</a> <a id="897" href="Categories.Adjoint.TwoSided.Compose.html#808" class="Bound">L</a> <a id="899" href="Categories.Adjoint.TwoSided.html#863" class="Record Operator">⊣⊢</a> <a id="902" href="Categories.Adjoint.TwoSided.Compose.html#826" class="Bound">R</a><a id="903" class="Symbol">)</a> <a id="905" class="Symbol">(</a><a id="906" href="Categories.Adjoint.TwoSided.Compose.html#906" class="Bound">L′⊣⊢R′</a> <a id="913" class="Symbol">:</a> <a id="915" href="Categories.Adjoint.TwoSided.Compose.html#844" class="Bound">L′</a> <a id="918" href="Categories.Adjoint.TwoSided.html#863" class="Record Operator">⊣⊢</a> <a id="921" href="Categories.Adjoint.TwoSided.Compose.html#863" class="Bound">R′</a><a id="923" class="Symbol">)</a> <a id="925" class="Symbol">→</a> <a id="927" class="Symbol">(</a><a id="928" href="Categories.Adjoint.TwoSided.Compose.html#844" class="Bound">L′</a> <a id="931" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="934" href="Categories.Adjoint.TwoSided.Compose.html#808" class="Bound">L</a><a id="935" class="Symbol">)</a> <a id="937" href="Categories.Adjoint.TwoSided.html#863" class="Record Operator">⊣⊢</a> <a id="940" class="Symbol">(</a><a id="941" href="Categories.Adjoint.TwoSided.Compose.html#826" class="Bound">R</a> <a id="943" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="946" href="Categories.Adjoint.TwoSided.Compose.html#863" class="Bound">R′</a><a id="948" class="Symbol">)</a>
<a id="950" href="Categories.Adjoint.TwoSided.Compose.html#722" class="Function Operator">_∘⊣⊢_</a> <a id="956" class="Symbol">{</a><a id="957" class="Argument">C</a> <a id="959" class="Symbol">=</a> <a id="961" href="Categories.Adjoint.TwoSided.Compose.html#961" class="Bound">C</a><a id="962" class="Symbol">}</a> <a id="964" class="Symbol">{</a><a id="965" href="Categories.Adjoint.TwoSided.Compose.html#965" class="Bound">D</a><a id="966" class="Symbol">}</a> <a id="968" class="Symbol">{</a><a id="969" href="Categories.Adjoint.TwoSided.Compose.html#969" class="Bound">E</a><a id="970" class="Symbol">}</a> <a id="972" class="Symbol">{</a><a id="973" href="Categories.Adjoint.TwoSided.Compose.html#973" class="Bound">L</a><a id="974" class="Symbol">}</a> <a id="976" class="Symbol">{</a><a id="977" href="Categories.Adjoint.TwoSided.Compose.html#977" class="Bound">R</a><a id="978" class="Symbol">}</a> <a id="980" class="Symbol">{</a><a id="981" href="Categories.Adjoint.TwoSided.Compose.html#981" class="Bound">L′</a><a id="983" class="Symbol">}</a> <a id="985" class="Symbol">{</a><a id="986" href="Categories.Adjoint.TwoSided.Compose.html#986" class="Bound">R′</a><a id="988" class="Symbol">}</a> <a id="990" href="Categories.Adjoint.TwoSided.Compose.html#990" class="Bound">L⊣⊢R</a> <a id="995" href="Categories.Adjoint.TwoSided.Compose.html#995" class="Bound">L′⊣⊢R′</a> <a id="1002" class="Symbol">=</a> <a id="1004" href="Categories.Adjoint.TwoSided.html#5870" class="Function">withZig</a> <a id="1012" class="Keyword">record</a>
<a id="1021" class="Symbol">{</a> <a id="1023" href="Categories.Adjoint.TwoSided.html#3118" class="Field">unit</a> <a id="1030" class="Symbol">=</a> <a id="1032" href="Categories.Adjoint.TwoSided.Compose.html#1629" class="Function">unit</a>
<a id="1039" class="Symbol">;</a> <a id="1041" href="Categories.Adjoint.TwoSided.html#3148" class="Field">counit</a> <a id="1048" class="Symbol">=</a> <a id="1050" href="Categories.Adjoint.TwoSided.Compose.html#3904" class="Function">counit</a>
<a id="1059" class="Symbol">;</a> <a id="1061" href="Categories.Adjoint.TwoSided.html#3564" class="Field">zig</a> <a id="1068" class="Symbol">=</a> <a id="1070" href="Categories.Adjoint.TwoSided.Compose.html#6327" class="Function">zig</a>
<a id="1076" class="Symbol">}</a>
<a id="1080" class="Keyword">where</a>
<a id="1088" class="Keyword">private</a>
<a id="1100" class="Keyword">module</a> <a id="1107" href="Categories.Adjoint.TwoSided.Compose.html#1107" class="Module">C</a> <a id="1111" class="Symbol">=</a> <a id="1113" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1122" href="Categories.Adjoint.TwoSided.Compose.html#961" class="Bound">C</a> <a id="1124" class="Keyword">using</a> <a id="1130" class="Symbol">(</a><a id="1131" href="Categories.Category.Core.html#656" class="Field Operator">_∘_</a><a id="1134" class="Symbol">;</a> <a id="1136" href="Categories.Category.Core.html#630" class="Field">id</a><a id="1138" class="Symbol">;</a> <a id="1140" href="Categories.Category.Core.html#715" class="Field">assoc</a><a id="1145" class="Symbol">;</a> <a id="1147" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a><a id="1156" class="Symbol">;</a> <a id="1158" class="Keyword">module</a> <a id="1165" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a><a id="1177" class="Symbol">)</a>
<a id="1183" class="Keyword">module</a> <a id="1190" href="Categories.Adjoint.TwoSided.Compose.html#1190" class="Module">D</a> <a id="1194" class="Symbol">=</a> <a id="1196" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1205" href="Categories.Adjoint.TwoSided.Compose.html#965" class="Bound">D</a> <a id="1207" class="Keyword">using</a> <a id="1213" class="Symbol">(</a><a id="1214" href="Categories.Category.Core.html#630" class="Field">id</a><a id="1216" class="Symbol">)</a>
<a id="1222" class="Keyword">module</a> <a id="1229" href="Categories.Adjoint.TwoSided.Compose.html#1229" class="Module">E</a> <a id="1233" class="Symbol">=</a> <a id="1235" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1244" href="Categories.Adjoint.TwoSided.Compose.html#969" class="Bound">E</a> <a id="1246" class="Keyword">using</a> <a id="1252" class="Symbol">(</a><a id="1253" href="Categories.Category.Core.html#656" class="Field Operator">_∘_</a><a id="1256" class="Symbol">;</a> <a id="1258" href="Categories.Category.Core.html#630" class="Field">id</a><a id="1260" class="Symbol">;</a> <a id="1262" href="Categories.Category.Core.html#595" class="Field Operator">_≈_</a><a id="1265" class="Symbol">;</a> <a id="1267" href="Categories.Category.Core.html#715" class="Field">assoc</a><a id="1272" class="Symbol">;</a> <a id="1274" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a><a id="1283" class="Symbol">;</a> <a id="1285" class="Keyword">module</a> <a id="1292" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a><a id="1304" class="Symbol">)</a>
<a id="1310" class="Keyword">module</a> <a id="1317" href="Categories.Adjoint.TwoSided.Compose.html#1317" class="Module">L</a> <a id="1321" class="Symbol">=</a> <a id="1323" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1331" href="Categories.Adjoint.TwoSided.Compose.html#973" class="Bound">L</a> <a id="1333" class="Keyword">using</a> <a id="1339" class="Symbol">(</a><a id="1340" href="Categories.Functor.Core.html#797" class="Function">₀</a><a id="1341" class="Symbol">;</a> <a id="1343" href="Categories.Functor.Core.html#806" class="Function">₁</a><a id="1344" class="Symbol">)</a>
<a id="1350" class="Keyword">module</a> <a id="1357" href="Categories.Adjoint.TwoSided.Compose.html#1357" class="Module">R</a> <a id="1361" class="Symbol">=</a> <a id="1363" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1371" href="Categories.Adjoint.TwoSided.Compose.html#977" class="Bound">R</a> <a id="1373" class="Keyword">using</a> <a id="1379" class="Symbol">(</a><a id="1380" href="Categories.Functor.Core.html#797" class="Function">₀</a><a id="1381" class="Symbol">;</a> <a id="1383" href="Categories.Functor.Core.html#806" class="Function">₁</a><a id="1384" class="Symbol">;</a> <a id="1386" href="Categories.Functor.Core.html#511" class="Field">identity</a><a id="1394" class="Symbol">)</a>
<a id="1400" class="Keyword">module</a> <a id="1407" href="Categories.Adjoint.TwoSided.Compose.html#1407" class="Module">L′</a> <a id="1411" class="Symbol">=</a> <a id="1413" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1421" href="Categories.Adjoint.TwoSided.Compose.html#981" class="Bound">L′</a> <a id="1424" class="Keyword">using</a> <a id="1430" class="Symbol">(</a><a id="1431" href="Categories.Functor.Core.html#797" class="Function">₀</a><a id="1432" class="Symbol">;</a> <a id="1434" href="Categories.Functor.Core.html#806" class="Function">₁</a><a id="1435" class="Symbol">;</a> <a id="1437" href="Categories.Functor.Core.html#511" class="Field">identity</a><a id="1445" class="Symbol">)</a>
<a id="1451" class="Keyword">module</a> <a id="1458" href="Categories.Adjoint.TwoSided.Compose.html#1458" class="Module">R′</a> <a id="1462" class="Symbol">=</a> <a id="1464" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1472" href="Categories.Adjoint.TwoSided.Compose.html#986" class="Bound">R′</a> <a id="1475" class="Keyword">using</a> <a id="1481" class="Symbol">(</a><a id="1482" href="Categories.Functor.Core.html#797" class="Function">₀</a><a id="1483" class="Symbol">;</a> <a id="1485" href="Categories.Functor.Core.html#806" class="Function">₁</a><a id="1486" class="Symbol">)</a>
<a id="1492" class="Keyword">module</a> <a id="1499" href="Categories.Adjoint.TwoSided.Compose.html#1499" class="Module">⊣⊢₁</a> <a id="1503" class="Symbol">=</a> <a id="1505" href="Categories.Adjoint.TwoSided.html#863" class="Module Operator">_⊣⊢_</a> <a id="1510" href="Categories.Adjoint.TwoSided.Compose.html#990" class="Bound">L⊣⊢R</a> <a id="1515" class="Keyword">using</a> <a id="1521" class="Symbol">(</a><a id="1522" href="Categories.Adjoint.TwoSided.html#1373" class="Field">zig</a><a id="1525" class="Symbol">;</a> <a id="1527" class="Keyword">module</a> <a id="1534" href="Categories.Adjoint.TwoSided.html#1022" class="Module">unit</a><a id="1538" class="Symbol">;</a> <a id="1540" class="Keyword">module</a> <a id="1547" href="Categories.Adjoint.TwoSided.html#1064" class="Module">counit</a><a id="1553" class="Symbol">)</a>
<a id="1559" class="Keyword">module</a> <a id="1566" href="Categories.Adjoint.TwoSided.Compose.html#1566" class="Module">⊣⊢₂</a> <a id="1570" class="Symbol">=</a> <a id="1572" href="Categories.Adjoint.TwoSided.html#863" class="Module Operator">_⊣⊢_</a> <a id="1577" href="Categories.Adjoint.TwoSided.Compose.html#995" class="Bound">L′⊣⊢R′</a> <a id="1584" class="Keyword">using</a> <a id="1590" class="Symbol">(</a><a id="1591" href="Categories.Adjoint.TwoSided.html#1373" class="Field">zig</a><a id="1594" class="Symbol">;</a> <a id="1596" class="Keyword">module</a> <a id="1603" href="Categories.Adjoint.TwoSided.html#1022" class="Module">unit</a><a id="1607" class="Symbol">;</a> <a id="1609" class="Keyword">module</a> <a id="1616" href="Categories.Adjoint.TwoSided.html#1064" class="Module">counit</a><a id="1622" class="Symbol">)</a>
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<a id="5897" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="5900" href="Categories.Morphism.Reasoning.Core.html#7259" class="Function">center</a> <a id="5907" class="Symbol">(</a><a id="5908" href="Categories.Functor.Properties.html#2426" class="Function Operator">[</a> <a id="5910" href="Categories.Adjoint.TwoSided.Compose.html#981" class="Bound">L′</a> <a id="5913" href="Categories.Functor.Properties.html#2426" class="Function Operator">]-resp-∘</a> <a id="5922" class="Symbol">(</a><a id="5923" href="Categories.Morphism.html#1612" class="Function">⊣⊢₁.counit.iso.isoʳ</a> <a id="5943" class="Symbol">(</a><a id="5944" href="Categories.Functor.Core.html#797" class="Function">R′.₀</a> <a id="5949" href="Categories.Adjoint.TwoSided.Compose.html#5226" class="Bound">e</a><a id="5950" class="Symbol">)))</a> <a id="5954" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="5966" href="Categories.NaturalTransformation.Core.html#783" class="Function">⊣⊢₂.counit.⇒.η</a> <a id="5981" href="Categories.Adjoint.TwoSided.Compose.html#5226" class="Bound">e</a> <a id="5983" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5985" href="Categories.Functor.Core.html#806" class="Function">L′.₁</a> <a id="5990" href="Categories.Category.Core.html#630" class="Function">D.id</a> <a id="5995" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5997" href="Categories.NaturalTransformation.Core.html#783" class="Function">⊣⊢₂.counit.⇐.η</a> <a id="6012" href="Categories.Adjoint.TwoSided.Compose.html#5226" class="Bound">e</a>
<a id="6026" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6029" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="6037" href="Categories.Morphism.Reasoning.Core.html#2948" class="Function">elimˡ</a> <a id="6043" href="Categories.Functor.Core.html#511" class="Function">L′.identity</a> <a id="6055" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="6067" href="Categories.NaturalTransformation.Core.html#783" class="Function">⊣⊢₂.counit.⇒.η</a> <a id="6082" href="Categories.Adjoint.TwoSided.Compose.html#5226" class="Bound">e</a> <a id="6084" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6086" href="Categories.NaturalTransformation.Core.html#783" class="Function">⊣⊢₂.counit.⇐.η</a> <a id="6101" href="Categories.Adjoint.TwoSided.Compose.html#5226" class="Bound">e</a>
<a id="6115" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6118" href="Categories.Morphism.html#1612" class="Function">⊣⊢₂.counit.iso.isoʳ</a> <a id="6138" href="Categories.Adjoint.TwoSided.Compose.html#5226" class="Bound">e</a> <a id="6140" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="6152" href="Categories.Category.Core.html#630" class="Function">id</a>
<a id="6167" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="6177" class="Symbol">}</a>
<a id="6185" class="Symbol">}</a>
<a id="6193" class="Keyword">where</a> <a id="6199" class="Keyword">open</a> <a id="6204" href="Categories.Adjoint.TwoSided.Compose.html#1229" class="Module">E</a>
<a id="6218" class="Keyword">open</a> <a id="6223" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="6248" class="Keyword">open</a> <a id="6253" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="6256" href="Categories.Adjoint.TwoSided.Compose.html#969" class="Bound">E</a>
<a id="6263" class="Keyword">module</a> <a id="6270" href="Categories.Adjoint.TwoSided.Compose.html#6270" class="Module">counit</a> <a id="6277" class="Symbol">=</a> <a id="6279" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Module">NaturalIsomorphism</a> <a id="6298" href="Categories.Adjoint.TwoSided.Compose.html#3904" class="Function">counit</a> <a id="6305" class="Keyword">using</a> <a id="6311" class="Symbol">(</a><a id="6312" class="Keyword">module</a> <a id="6319" href="Categories.NaturalTransformation.NaturalIsomorphism.html#969" class="Module">⇒</a><a id="6320" class="Symbol">)</a>
<a id="6327" href="Categories.Adjoint.TwoSided.Compose.html#6327" class="Function">zig</a> <a id="6331" class="Symbol">:</a> <a id="6333" class="Symbol">∀</a> <a id="6335" class="Symbol">{</a><a id="6336" href="Categories.Adjoint.TwoSided.Compose.html#6336" class="Bound">c</a><a id="6337" class="Symbol">}</a> <a id="6339" class="Symbol">→</a> <a id="6341" href="Categories.NaturalTransformation.Core.html#783" class="Function">counit.⇒.η</a> <a id="6352" class="Symbol">(</a><a id="6353" href="Categories.Functor.Core.html#797" class="Function">L′.₀</a> <a id="6358" class="Symbol">(</a><a id="6359" href="Categories.Functor.Core.html#797" class="Function">L.₀</a> <a id="6363" href="Categories.Adjoint.TwoSided.Compose.html#6336" class="Bound">c</a><a id="6364" class="Symbol">))</a> <a id="6367" href="Categories.Category.Core.html#656" class="Function Operator">E.∘</a> <a id="6371" href="Categories.Functor.Core.html#806" class="Function">L′.₁</a> <a id="6376" class="Symbol">(</a><a id="6377" href="Categories.Functor.Core.html#806" class="Function">L.₁</a> <a id="6381" class="Symbol">(</a><a id="6382" href="Categories.NaturalTransformation.Core.html#783" class="Function">unit.⇒.η</a> <a id="6391" href="Categories.Adjoint.TwoSided.Compose.html#6336" class="Bound">c</a><a id="6392" class="Symbol">))</a> <a id="6395" href="Categories.Category.Core.html#595" class="Function Operator">E.≈</a> <a id="6399" href="Categories.Category.Core.html#630" class="Function">E.id</a>
<a id="6408" href="Categories.Adjoint.TwoSided.Compose.html#6327" class="Function">zig</a> <a id="6412" class="Symbol">{</a><a id="6413" href="Categories.Adjoint.TwoSided.Compose.html#6413" class="Bound">c</a><a id="6414" class="Symbol">}</a> <a id="6416" class="Symbol">=</a> <a id="6418" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="6430" href="Categories.NaturalTransformation.Core.html#783" class="Function">counit.⇒.η</a> <a id="6441" class="Symbol">(</a><a id="6442" href="Categories.Functor.Core.html#797" class="Function">L′.₀</a> <a id="6447" class="Symbol">(</a><a id="6448" href="Categories.Functor.Core.html#797" class="Function">L.₀</a> <a id="6452" href="Categories.Adjoint.TwoSided.Compose.html#6413" class="Bound">c</a><a id="6453" class="Symbol">))</a> <a id="6456" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6458" href="Categories.Functor.Core.html#806" class="Function">L′.₁</a> <a id="6463" class="Symbol">(</a><a id="6464" href="Categories.Functor.Core.html#806" class="Function">L.₁</a> <a id="6468" class="Symbol">(</a><a id="6469" href="Categories.NaturalTransformation.Core.html#783" class="Function">unit.⇒.η</a> <a id="6478" href="Categories.Adjoint.TwoSided.Compose.html#6413" class="Bound">c</a><a id="6479" class="Symbol">))</a>
<a id="6490" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6493" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="6501" href="Categories.Functor.Core.html#565" class="Field">Functor.homomorphism</a> <a id="6522" class="Symbol">(</a><a id="6523" href="Categories.Adjoint.TwoSided.Compose.html#981" class="Bound">L′</a> <a id="6526" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="6529" href="Categories.Adjoint.TwoSided.Compose.html#973" class="Bound">L</a><a id="6530" class="Symbol">)</a> <a id="6532" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="6540" href="Categories.NaturalTransformation.Core.html#783" class="Function">counit.⇒.η</a> <a id="6551" class="Symbol">(</a><a id="6552" href="Categories.Functor.Core.html#797" class="Function">L′.₀</a> <a id="6557" class="Symbol">(</a><a id="6558" href="Categories.Functor.Core.html#797" class="Function">L.₀</a> <a id="6562" href="Categories.Adjoint.TwoSided.Compose.html#6413" class="Bound">c</a><a id="6563" class="Symbol">))</a> <a id="6566" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6568" href="Categories.Functor.Core.html#806" class="Function">L′.₁</a> <a id="6573" class="Symbol">(</a><a id="6574" href="Categories.Functor.Core.html#806" class="Function">L.₁</a> <a id="6578" class="Symbol">(</a><a id="6579" href="Categories.Functor.Core.html#806" class="Function">R.₁</a> <a id="6583" class="Symbol">(</a><a id="6584" href="Categories.NaturalTransformation.Core.html#783" class="Function">⊣⊢₂.unit.⇒.η</a> <a id="6597" class="Symbol">(</a><a id="6598" href="Categories.Functor.Core.html#797" class="Function">L.₀</a> <a id="6602" href="Categories.Adjoint.TwoSided.Compose.html#6413" class="Bound">c</a><a id="6603" class="Symbol">))))</a> <a id="6608" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6610" href="Categories.Functor.Core.html#806" class="Function">L′.₁</a> <a id="6615" class="Symbol">(</a><a id="6616" href="Categories.Functor.Core.html#806" class="Function">L.₁</a> <a id="6620" class="Symbol">(</a><a id="6621" href="Categories.NaturalTransformation.Core.html#783" class="Function">⊣⊢₁.unit.⇒.η</a> <a id="6634" href="Categories.Adjoint.TwoSided.Compose.html#6413" class="Bound">c</a><a id="6635" class="Symbol">))</a>
<a id="6646" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6649" href="Categories.Morphism.Reasoning.Core.html#7259" class="Function">center</a> <a id="6656" class="Symbol">(</a><a id="6657" href="Categories.Functor.Properties.html#2676" class="Function Operator">[</a> <a id="6659" href="Categories.Adjoint.TwoSided.Compose.html#981" class="Bound">L′</a> <a id="6662" href="Categories.Functor.Properties.html#2676" class="Function Operator">]-resp-square</a> <a id="6676" class="Symbol">(</a><a id="6677" href="Categories.NaturalTransformation.Core.html#827" class="Function">⊣⊢₁.counit.⇒.commute</a> <a id="6698" class="Symbol">(</a><a id="6699" href="Categories.NaturalTransformation.Core.html#783" class="Function">⊣⊢₂.unit.⇒.η</a> <a id="6712" class="Symbol">(</a><a id="6713" href="Categories.Functor.Core.html#797" class="Function">L.₀</a> <a id="6717" href="Categories.Adjoint.TwoSided.Compose.html#6413" class="Bound">c</a><a id="6718" class="Symbol">))))</a> <a id="6723" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="6731" href="Categories.NaturalTransformation.Core.html#783" class="Function">⊣⊢₂.counit.⇒.η</a> <a id="6746" class="Symbol">(</a><a id="6747" href="Categories.Functor.Core.html#797" class="Function">L′.₀</a> <a id="6752" class="Symbol">(</a><a id="6753" href="Categories.Functor.Core.html#797" class="Function">L.₀</a> <a id="6757" href="Categories.Adjoint.TwoSided.Compose.html#6413" class="Bound">c</a><a id="6758" class="Symbol">))</a> <a id="6761" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6763" class="Symbol">(</a><a id="6764" href="Categories.Functor.Core.html#806" class="Function">L′.₁</a> <a id="6769" class="Symbol">(</a><a id="6770" href="Categories.NaturalTransformation.Core.html#783" class="Function">⊣⊢₂.unit.⇒.η</a> <a id="6783" class="Symbol">(</a><a id="6784" href="Categories.Functor.Core.html#797" class="Function">L.₀</a> <a id="6788" href="Categories.Adjoint.TwoSided.Compose.html#6413" class="Bound">c</a><a id="6789" class="Symbol">))</a> <a id="6792" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6794" href="Categories.Functor.Core.html#806" class="Function">L′.₁</a> <a id="6799" class="Symbol">(</a><a id="6800" href="Categories.NaturalTransformation.Core.html#783" class="Function">⊣⊢₁.counit.⇒.η</a> <a id="6815" class="Symbol">(</a><a id="6816" href="Categories.Functor.Core.html#797" class="Function">L.₀</a> <a id="6820" href="Categories.Adjoint.TwoSided.Compose.html#6413" class="Bound">c</a><a id="6821" class="Symbol">)))</a> <a id="6825" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6827" href="Categories.Functor.Core.html#806" class="Function">L′.₁</a> <a id="6832" class="Symbol">(</a><a id="6833" href="Categories.Functor.Core.html#806" class="Function">L.₁</a> <a id="6837" class="Symbol">(</a><a id="6838" href="Categories.NaturalTransformation.Core.html#783" class="Function">⊣⊢₁.unit.⇒.η</a> <a id="6851" href="Categories.Adjoint.TwoSided.Compose.html#6413" class="Bound">c</a><a id="6852" class="Symbol">))</a>
<a id="6863" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6866" href="Categories.Morphism.Reasoning.Core.html#7730" class="Function">pull-first</a> <a id="6877" href="Categories.Adjoint.TwoSided.html#1373" class="Field">⊣⊢₂.zig</a> <a id="6885" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="6893" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6896" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6898" href="Categories.Functor.Core.html#806" class="Function">L′.₁</a> <a id="6903" class="Symbol">(</a><a id="6904" href="Categories.NaturalTransformation.Core.html#783" class="Function">⊣⊢₁.counit.⇒.η</a> <a id="6919" class="Symbol">(</a><a id="6920" href="Categories.Functor.Core.html#797" class="Function">L.₀</a> <a id="6924" href="Categories.Adjoint.TwoSided.Compose.html#6413" class="Bound">c</a><a id="6925" class="Symbol">))</a> <a id="6928" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6930" href="Categories.Functor.Core.html#806" class="Function">L′.₁</a> <a id="6935" class="Symbol">(</a><a id="6936" href="Categories.Functor.Core.html#806" class="Function">L.₁</a> <a id="6940" class="Symbol">(</a><a id="6941" href="Categories.NaturalTransformation.Core.html#783" class="Function">⊣⊢₁.unit.⇒.η</a> <a id="6954" href="Categories.Adjoint.TwoSided.Compose.html#6413" class="Bound">c</a><a id="6955" class="Symbol">))</a>
<a id="6966" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6969" href="Categories.Morphism.Reasoning.Core.html#2786" class="Function">elimʳ</a> <a id="6975" class="Symbol">((</a><a id="6977" href="Categories.Functor.Properties.html#2426" class="Function Operator">[</a> <a id="6979" href="Categories.Adjoint.TwoSided.Compose.html#981" class="Bound">L′</a> <a id="6982" href="Categories.Functor.Properties.html#2426" class="Function Operator">]-resp-∘</a> <a id="6991" href="Categories.Adjoint.TwoSided.html#1373" class="Function">⊣⊢₁.zig</a><a id="6998" class="Symbol">)</a> <a id="7000" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="7002" href="Categories.Functor.Core.html#511" class="Function">L′.identity</a><a id="7013" class="Symbol">)</a> <a id="7015" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="7023" href="Categories.Category.Core.html#630" class="Function">id</a>
<a id="7034" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="7042" class="Keyword">where</a> <a id="7048" class="Keyword">open</a> <a id="7053" href="Categories.Adjoint.TwoSided.Compose.html#1229" class="Module">E</a>
<a id="7067" class="Keyword">open</a> <a id="7072" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="7097" class="Keyword">open</a> <a id="7102" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="7105" href="Categories.Adjoint.TwoSided.Compose.html#969" class="Bound">E</a>
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