bsc-leon-vatthauer/agda/bsc-thesis/Categories.Category.Extensive.html

<!DOCTYPE HTML>
<html><head><meta charset="utf-8"><title>Categories.Category.Extensive</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.Category.Extensive.html" class="Module">Categories.Category.Extensive</a> <a id="74" class="Keyword">where</a>

<a id="81" class="Comment">-- https://ncatlab.org/nlab/show/extensive+category</a>

<a id="134" class="Keyword">open</a> <a id="139" class="Keyword">import</a> <a id="146" href="Level.html" class="Module">Level</a>
<a id="152" class="Keyword">open</a> <a id="157" class="Keyword">import</a> <a id="164" href="Function.html" class="Module">Function</a> <a id="173" class="Keyword">using</a> <a id="179" class="Symbol">(</a><a id="180" href="Function.Base.html#1974" class="Function Operator">_$_</a><a id="183" class="Symbol">)</a>

<a id="186" class="Keyword">open</a> <a id="191" class="Keyword">import</a> <a id="198" href="Categories.Category.Core.html" class="Module">Categories.Category.Core</a> <a id="223" class="Keyword">using</a> <a id="229" class="Symbol">(</a><a id="230" href="Categories.Category.Core.html#442" class="Record">Category</a><a id="238" class="Symbol">)</a>
<a id="240" class="Keyword">open</a> <a id="245" class="Keyword">import</a> <a id="252" href="Categories.Diagram.Pullback.html" class="Module">Categories.Diagram.Pullback</a> <a id="280" class="Keyword">using</a> <a id="286" class="Symbol">(</a><a id="287" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a><a id="295" class="Symbol">;</a> <a id="297" href="Categories.Diagram.Pullback.html#747" class="Record">IsPullback</a><a id="307" class="Symbol">;</a> <a id="309" href="Categories.Diagram.Pullback.html#2216" class="Function">up-to-iso</a><a id="318" class="Symbol">)</a>
<a id="320" class="Keyword">open</a> <a id="325" class="Keyword">import</a> <a id="332" href="Categories.Diagram.Pullback.Properties.html" class="Module">Categories.Diagram.Pullback.Properties</a> <a id="371" class="Keyword">using</a> <a id="377" class="Symbol">(</a><a id="378" class="Keyword">module</a> <a id="385" href="Categories.Diagram.Pullback.Properties.html#6429" class="Module">IsoPb</a><a id="390" class="Symbol">)</a>
<a id="392" class="Keyword">open</a> <a id="397" class="Keyword">import</a> <a id="404" href="Categories.Category.Cocartesian.html" class="Module">Categories.Category.Cocartesian</a> <a id="436" class="Keyword">using</a> <a id="442" class="Symbol">(</a><a id="443" href="Categories.Category.Cocartesian.html#3385" class="Record">Cocartesian</a><a id="454" class="Symbol">)</a>
<a id="456" class="Keyword">open</a> <a id="461" class="Keyword">import</a> <a id="468" href="Categories.Object.Coproduct.html" class="Module">Categories.Object.Coproduct</a> <a id="496" class="Keyword">using</a> <a id="502" class="Symbol">(</a><a id="503" href="Categories.Object.Coproduct.html#1139" class="Record">IsCoproduct</a><a id="514" class="Symbol">)</a>
<a id="516" class="Keyword">open</a> <a id="521" class="Keyword">import</a> <a id="528" href="Categories.Morphism.html" class="Module">Categories.Morphism</a>
<a id="548" class="Keyword">import</a> <a id="555" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="585" class="Symbol">as</a> <a id="588" class="Module">MR</a>

<a id="592" class="Keyword">record</a> <a id="Extensive"></a><a id="599" href="Categories.Category.Extensive.html#599" class="Record">Extensive</a> <a id="609" class="Symbol">{</a><a id="610" href="Categories.Category.Extensive.html#610" class="Bound">o</a> <a id="612" href="Categories.Category.Extensive.html#612" class="Bound">ℓ</a> <a id="614" href="Categories.Category.Extensive.html#614" class="Bound">e</a> <a id="616" class="Symbol">:</a> <a id="618" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="623" class="Symbol">}</a> <a id="625" class="Symbol">(</a><a id="626" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="628" class="Symbol">:</a> <a id="630" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="639" href="Categories.Category.Extensive.html#610" class="Bound">o</a> <a id="641" href="Categories.Category.Extensive.html#612" class="Bound">ℓ</a> <a id="643" href="Categories.Category.Extensive.html#614" class="Bound">e</a><a id="644" class="Symbol">)</a> <a id="646" class="Symbol">:</a> <a id="648" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="652" class="Symbol">(</a><a id="653" href="Agda.Primitive.html#931" class="Primitive">suc</a> <a id="657" class="Symbol">(</a><a id="658" href="Categories.Category.Extensive.html#610" class="Bound">o</a> <a id="660" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="662" href="Categories.Category.Extensive.html#612" class="Bound">ℓ</a> <a id="664" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="666" href="Categories.Category.Extensive.html#614" class="Bound">e</a><a id="667" class="Symbol">))</a> <a id="670" class="Keyword">where</a>
  <a id="678" class="Keyword">open</a> <a id="683" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="692" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a>
  <a id="696" class="Keyword">open</a> <a id="701" href="Categories.Diagram.Pullback.html#2029" class="Module">Pullback</a> <a id="710" class="Keyword">using</a> <a id="716" class="Symbol">(</a><a id="717" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a><a id="719" class="Symbol">)</a>

  <a id="724" class="Keyword">field</a>
    <a id="Extensive.cocartesian"></a><a id="734" href="Categories.Category.Extensive.html#734" class="Field">cocartesian</a> <a id="746" class="Symbol">:</a> <a id="748" href="Categories.Category.Cocartesian.html#3385" class="Record">Cocartesian</a> <a id="760" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a>

  <a id="765" class="Keyword">module</a> <a id="Extensive.CC"></a><a id="772" href="Categories.Category.Extensive.html#772" class="Module">CC</a> <a id="775" class="Symbol">=</a> <a id="777" href="Categories.Category.Cocartesian.html#3385" class="Module">Cocartesian</a> <a id="789" href="Categories.Category.Extensive.html#734" class="Field">cocartesian</a>
  <a id="803" class="Keyword">open</a> <a id="808" href="Categories.Category.Extensive.html#772" class="Module">CC</a> <a id="811" class="Keyword">using</a> <a id="817" class="Symbol">(</a><a id="818" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">_+_</a><a id="821" class="Symbol">;</a> <a id="823" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="825" class="Symbol">;</a> <a id="827" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="829" class="Symbol">;</a> <a id="831" href="Categories.Category.Cocartesian.html#3616" class="Function">¡</a><a id="832" class="Symbol">)</a>

  <a id="837" class="Keyword">field</a>
    <a id="Extensive.pullback₁"></a><a id="847" href="Categories.Category.Extensive.html#847" class="Field">pullback₁</a> <a id="857" class="Symbol">:</a> <a id="859" class="Symbol">{</a><a id="860" href="Categories.Category.Extensive.html#860" class="Bound">A</a> <a id="862" href="Categories.Category.Extensive.html#862" class="Bound">B</a> <a id="864" href="Categories.Category.Extensive.html#864" class="Bound">C</a> <a id="866" class="Symbol">:</a> <a id="868" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="871" class="Symbol">}</a> <a id="873" class="Symbol">(</a><a id="874" href="Categories.Category.Extensive.html#874" class="Bound">f</a> <a id="876" class="Symbol">:</a> <a id="878" href="Categories.Category.Extensive.html#860" class="Bound">A</a> <a id="880" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="882" href="Categories.Category.Extensive.html#862" class="Bound">B</a> <a id="884" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="886" href="Categories.Category.Extensive.html#864" class="Bound">C</a><a id="887" class="Symbol">)</a> <a id="889" class="Symbol">→</a> <a id="891" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="900" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="902" href="Categories.Category.Extensive.html#874" class="Bound">f</a> <a id="904" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a>
    <a id="Extensive.pullback₂"></a><a id="911" href="Categories.Category.Extensive.html#911" class="Field">pullback₂</a> <a id="921" class="Symbol">:</a> <a id="923" class="Symbol">{</a><a id="924" href="Categories.Category.Extensive.html#924" class="Bound">A</a> <a id="926" href="Categories.Category.Extensive.html#926" class="Bound">B</a> <a id="928" href="Categories.Category.Extensive.html#928" class="Bound">C</a> <a id="930" class="Symbol">:</a> <a id="932" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="935" class="Symbol">}</a> <a id="937" class="Symbol">(</a><a id="938" href="Categories.Category.Extensive.html#938" class="Bound">f</a> <a id="940" class="Symbol">:</a> <a id="942" href="Categories.Category.Extensive.html#924" class="Bound">A</a> <a id="944" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="946" href="Categories.Category.Extensive.html#926" class="Bound">B</a> <a id="948" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="950" href="Categories.Category.Extensive.html#928" class="Bound">C</a><a id="951" class="Symbol">)</a> <a id="953" class="Symbol">→</a> <a id="955" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="964" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="966" href="Categories.Category.Extensive.html#938" class="Bound">f</a> <a id="968" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a>
    <a id="Extensive.pullback-of-cp-is-cp"></a><a id="975" href="Categories.Category.Extensive.html#975" class="Field">pullback-of-cp-is-cp</a> <a id="996" class="Symbol">:</a> <a id="998" class="Symbol">{</a><a id="999" href="Categories.Category.Extensive.html#999" class="Bound">A</a> <a id="1001" href="Categories.Category.Extensive.html#1001" class="Bound">B</a> <a id="1003" href="Categories.Category.Extensive.html#1003" class="Bound">C</a> <a id="1005" class="Symbol">:</a> <a id="1007" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="1010" class="Symbol">}</a> <a id="1012" class="Symbol">(</a><a id="1013" href="Categories.Category.Extensive.html#1013" class="Bound">f</a> <a id="1015" class="Symbol">:</a> <a id="1017" href="Categories.Category.Extensive.html#999" class="Bound">A</a> <a id="1019" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1021" href="Categories.Category.Extensive.html#1001" class="Bound">B</a> <a id="1023" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1025" href="Categories.Category.Extensive.html#1003" class="Bound">C</a><a id="1026" class="Symbol">)</a> <a id="1028" class="Symbol">→</a> <a id="1030" href="Categories.Object.Coproduct.html#1139" class="Record">IsCoproduct</a> <a id="1042" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="1044" class="Symbol">(</a><a id="1045" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="1048" class="Symbol">(</a><a id="1049" href="Categories.Category.Extensive.html#847" class="Field">pullback₁</a> <a id="1059" href="Categories.Category.Extensive.html#1013" class="Bound">f</a><a id="1060" class="Symbol">))</a> <a id="1063" class="Symbol">(</a><a id="1064" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="1067" class="Symbol">(</a><a id="1068" href="Categories.Category.Extensive.html#911" class="Field">pullback₂</a> <a id="1078" href="Categories.Category.Extensive.html#1013" class="Bound">f</a><a id="1079" class="Symbol">))</a>

    <a id="Extensive.pullback₁-is-mono"></a><a id="1091" href="Categories.Category.Extensive.html#1091" class="Field">pullback₁-is-mono</a> <a id="1109" class="Symbol">:</a> <a id="1111" class="Symbol">∀</a> <a id="1113" class="Symbol">{</a><a id="1114" href="Categories.Category.Extensive.html#1114" class="Bound">A</a> <a id="1116" href="Categories.Category.Extensive.html#1116" class="Bound">B</a> <a id="1118" class="Symbol">:</a> <a id="1120" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="1123" class="Symbol">}</a> <a id="1125" class="Symbol">→</a> <a id="1127" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="1132" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="1134" class="Symbol">(</a><a id="1135" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1138" class="Symbol">{</a><a id="1139" class="Argument">A</a> <a id="1141" class="Symbol">=</a> <a id="1143" href="Categories.Category.Extensive.html#1114" class="Bound">A</a><a id="1144" class="Symbol">}{</a><a id="1146" class="Argument">B</a> <a id="1148" class="Symbol">=</a> <a id="1150" href="Categories.Category.Extensive.html#1116" class="Bound">B</a><a id="1151" class="Symbol">})</a>
    <a id="Extensive.pullback₂-is-mono"></a><a id="1158" href="Categories.Category.Extensive.html#1158" class="Field">pullback₂-is-mono</a> <a id="1176" class="Symbol">:</a> <a id="1178" class="Symbol">∀</a> <a id="1180" class="Symbol">{</a><a id="1181" href="Categories.Category.Extensive.html#1181" class="Bound">A</a> <a id="1183" href="Categories.Category.Extensive.html#1183" class="Bound">B</a> <a id="1185" class="Symbol">:</a> <a id="1187" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="1190" class="Symbol">}</a> <a id="1192" class="Symbol">→</a> <a id="1194" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="1199" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="1201" class="Symbol">(</a><a id="1202" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1205" class="Symbol">{</a><a id="1206" class="Argument">A</a> <a id="1208" class="Symbol">=</a> <a id="1210" href="Categories.Category.Extensive.html#1181" class="Bound">A</a><a id="1211" class="Symbol">}{</a><a id="1213" class="Argument">B</a> <a id="1215" class="Symbol">=</a> <a id="1217" href="Categories.Category.Extensive.html#1183" class="Bound">B</a><a id="1218" class="Symbol">})</a>

    <a id="Extensive.disjoint"></a><a id="1226" href="Categories.Category.Extensive.html#1226" class="Field">disjoint</a> <a id="1235" class="Symbol">:</a> <a id="1237" class="Symbol">∀</a> <a id="1239" class="Symbol">{</a><a id="1240" href="Categories.Category.Extensive.html#1240" class="Bound">A</a> <a id="1242" href="Categories.Category.Extensive.html#1242" class="Bound">B</a> <a id="1244" class="Symbol">:</a> <a id="1246" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="1249" class="Symbol">}</a> <a id="1251" class="Symbol">→</a> <a id="1253" href="Categories.Diagram.Pullback.html#747" class="Record">IsPullback</a> <a id="1264" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="1266" href="Categories.Category.Cocartesian.html#3616" class="Function">¡</a> <a id="1268" href="Categories.Category.Cocartesian.html#3616" class="Function">¡</a> <a id="1270" class="Symbol">(</a><a id="1271" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1274" class="Symbol">{</a><a id="1275" class="Argument">A</a> <a id="1277" class="Symbol">=</a> <a id="1279" href="Categories.Category.Extensive.html#1240" class="Bound">A</a><a id="1280" class="Symbol">}{</a><a id="1282" class="Argument">B</a> <a id="1284" class="Symbol">=</a> <a id="1286" href="Categories.Category.Extensive.html#1242" class="Bound">B</a><a id="1287" class="Symbol">})</a> <a id="1290" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a>

  <a id="1296" class="Comment">-- a version with non-canonical pullbacks</a>
  <a id="1340" class="Keyword">module</a> <a id="1347" href="Categories.Category.Extensive.html#1347" class="Module">_</a> <a id="1349" class="Symbol">{</a><a id="1350" href="Categories.Category.Extensive.html#1350" class="Bound">A</a> <a id="1352" href="Categories.Category.Extensive.html#1352" class="Bound">B</a> <a id="1354" href="Categories.Category.Extensive.html#1354" class="Bound">C</a> <a id="1356" class="Symbol">:</a> <a id="1358" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="1361" class="Symbol">}</a> <a id="1363" class="Symbol">{</a><a id="1364" href="Categories.Category.Extensive.html#1364" class="Bound">f</a> <a id="1366" class="Symbol">:</a> <a id="1368" href="Categories.Category.Extensive.html#1350" class="Bound">A</a> <a id="1370" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1372" href="Categories.Category.Extensive.html#1352" class="Bound">B</a> <a id="1374" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1376" href="Categories.Category.Extensive.html#1354" class="Bound">C</a><a id="1377" class="Symbol">}</a> <a id="1379" class="Symbol">(</a><a id="1380" href="Categories.Category.Extensive.html#1380" class="Bound">pb₁</a> <a id="1384" class="Symbol">:</a> <a id="1386" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="1395" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="1397" href="Categories.Category.Extensive.html#1364" class="Bound">f</a> <a id="1399" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="1401" class="Symbol">)</a> <a id="1403" class="Symbol">(</a><a id="1404" href="Categories.Category.Extensive.html#1404" class="Bound">pb₂</a> <a id="1408" class="Symbol">:</a> <a id="1410" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="1419" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="1421" href="Categories.Category.Extensive.html#1364" class="Bound">f</a> <a id="1423" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="1425" class="Symbol">)</a> <a id="1427" class="Keyword">where</a>
      <a id="1439" class="Keyword">private</a>
        <a id="1456" class="Keyword">open</a> <a id="1461" href="Categories.Object.Coproduct.html#1139" class="Module">IsCoproduct</a> <a id="1473" class="Symbol">(</a><a id="1474" href="Categories.Category.Extensive.html#975" class="Field">pullback-of-cp-is-cp</a> <a id="1495" href="Categories.Category.Extensive.html#1364" class="Bound">f</a><a id="1496" class="Symbol">)</a>
        <a id="1506" class="Keyword">open</a> <a id="1511" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a><a id="1523" class="Symbol">;</a> <a id="1525" class="Keyword">open</a> <a id="1530" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="1533" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a>

        <a id="1550" class="Keyword">open</a> <a id="1555" href="Categories.Diagram.Pullback.Properties.html#6429" class="Module">IsoPb</a> <a id="1561" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="1563" class="Symbol">(</a><a id="1564" href="Categories.Category.Extensive.html#847" class="Field">pullback₁</a> <a id="1574" href="Categories.Category.Extensive.html#1364" class="Bound">f</a><a id="1575" class="Symbol">)</a> <a id="1577" href="Categories.Category.Extensive.html#1380" class="Bound">pb₁</a> <a id="1581" class="Keyword">renaming</a> <a id="1590" class="Symbol">(</a><a id="1591" href="Categories.Diagram.Pullback.Properties.html#6654" class="Function">P₀⇒P₁</a> <a id="1597" class="Symbol">to</a> <a id="1600" class="Function">pb₁-to-can</a><a id="1610" class="Symbol">;</a> <a id="1612" href="Categories.Diagram.Pullback.Properties.html#6708" class="Function">p₁-≈</a> <a id="1617" class="Symbol">to</a> <a id="1620" class="Function">p₁-≈₁</a><a id="1625" class="Symbol">)</a>
        <a id="1635" class="Keyword">open</a> <a id="1640" href="Categories.Diagram.Pullback.Properties.html#6429" class="Module">IsoPb</a> <a id="1646" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="1648" class="Symbol">(</a><a id="1649" href="Categories.Category.Extensive.html#911" class="Field">pullback₂</a> <a id="1659" href="Categories.Category.Extensive.html#1364" class="Bound">f</a><a id="1660" class="Symbol">)</a> <a id="1662" href="Categories.Category.Extensive.html#1404" class="Bound">pb₂</a> <a id="1666" class="Keyword">renaming</a> <a id="1675" class="Symbol">(</a><a id="1676" href="Categories.Diagram.Pullback.Properties.html#6654" class="Function">P₀⇒P₁</a> <a id="1682" class="Symbol">to</a> <a id="1685" class="Function">pb₂-to-can</a><a id="1695" class="Symbol">;</a> <a id="1697" href="Categories.Diagram.Pullback.Properties.html#6708" class="Function">p₁-≈</a> <a id="1702" class="Symbol">to</a> <a id="1705" class="Function">p₁-≈₂</a><a id="1710" class="Symbol">)</a>

        <a id="1721" href="Categories.Category.Extensive.html#1721" class="Function">can-to-pb₁</a> <a id="1732" class="Symbol">=</a> <a id="1734" href="Categories.Morphism.html#2006" class="Field">_≅_.from</a> <a id="1743" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="1745" href="Categories.Diagram.Pullback.html#2216" class="Function">up-to-iso</a> <a id="1755" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="1757" href="Categories.Category.Extensive.html#1380" class="Bound">pb₁</a> <a id="1761" class="Symbol">(</a><a id="1762" href="Categories.Category.Extensive.html#847" class="Field">pullback₁</a> <a id="1772" href="Categories.Category.Extensive.html#1364" class="Bound">f</a><a id="1773" class="Symbol">)</a>
        <a id="1783" href="Categories.Category.Extensive.html#1783" class="Function">can-to-pb₂</a> <a id="1794" class="Symbol">=</a> <a id="1796" href="Categories.Morphism.html#2006" class="Field">_≅_.from</a> <a id="1805" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="1807" href="Categories.Diagram.Pullback.html#2216" class="Function">up-to-iso</a> <a id="1817" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="1819" href="Categories.Category.Extensive.html#1404" class="Bound">pb₂</a> <a id="1823" class="Symbol">(</a><a id="1824" href="Categories.Category.Extensive.html#911" class="Field">pullback₂</a> <a id="1834" href="Categories.Category.Extensive.html#1364" class="Bound">f</a><a id="1835" class="Symbol">)</a>

      <a id="1844" href="Categories.Category.Extensive.html#1844" class="Function">pullback-of-cp-is-cp&#39;</a> <a id="1866" class="Symbol">:</a> <a id="1868" href="Categories.Object.Coproduct.html#1139" class="Record">IsCoproduct</a> <a id="1880" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="1882" class="Symbol">(</a><a id="1883" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="1886" href="Categories.Category.Extensive.html#1380" class="Bound">pb₁</a><a id="1889" class="Symbol">)</a> <a id="1891" class="Symbol">(</a><a id="1892" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="1895" href="Categories.Category.Extensive.html#1404" class="Bound">pb₂</a><a id="1898" class="Symbol">)</a>

      <a id="1913" href="Categories.Object.Coproduct.html#1233" class="Field Operator">IsCoproduct.[_,_]</a> <a id="1931" href="Categories.Category.Extensive.html#1844" class="Function">pullback-of-cp-is-cp&#39;</a> <a id="1953" href="Categories.Category.Extensive.html#1953" class="Bound">g</a> <a id="1955" href="Categories.Category.Extensive.html#1955" class="Bound">h</a>  <a id="1958" class="Symbol">=</a> <a id="1960" href="Categories.Object.Coproduct.html#1233" class="Function Operator">[</a> <a id="1962" href="Categories.Category.Extensive.html#1953" class="Bound">g</a> <a id="1964" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1966" href="Categories.Category.Extensive.html#1600" class="Function">pb₁-to-can</a> <a id="1977" href="Categories.Object.Coproduct.html#1233" class="Function Operator">,</a> <a id="1979" href="Categories.Category.Extensive.html#1955" class="Bound">h</a> <a id="1981" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1983" href="Categories.Category.Extensive.html#1685" class="Function">pb₂-to-can</a> <a id="1994" href="Categories.Object.Coproduct.html#1233" class="Function Operator">]</a>
      <a id="2002" href="Categories.Object.Coproduct.html#1270" class="Field">IsCoproduct.inject₁</a> <a id="2022" href="Categories.Category.Extensive.html#1844" class="Function">pullback-of-cp-is-cp&#39;</a> <a id="2044" class="Symbol">{_}{</a><a id="2048" href="Categories.Category.Extensive.html#2048" class="Bound">g</a><a id="2049" class="Symbol">}{</a><a id="2051" href="Categories.Category.Extensive.html#2051" class="Bound">h</a><a id="2052" class="Symbol">}</a>  <a id="2055" class="Symbol">=</a> <a id="2057" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
         <a id="2072" href="Categories.Object.Coproduct.html#1233" class="Function Operator">[</a> <a id="2074" href="Categories.Category.Extensive.html#2048" class="Bound">g</a> <a id="2076" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2078" href="Categories.Category.Extensive.html#1600" class="Function">pb₁-to-can</a> <a id="2089" href="Categories.Object.Coproduct.html#1233" class="Function Operator">,</a> <a id="2091" href="Categories.Category.Extensive.html#2051" class="Bound">h</a> <a id="2093" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2095" href="Categories.Category.Extensive.html#1685" class="Function">pb₂-to-can</a> <a id="2106" href="Categories.Object.Coproduct.html#1233" class="Function Operator">]</a> <a id="2108" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2110" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="2113" href="Categories.Category.Extensive.html#1380" class="Bound">pb₁</a>                               <a id="2147" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="2151" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="2159" href="Categories.Morphism.Reasoning.Core.html#6665" class="Function">cancelʳ</a> <a id="2167" class="Symbol">(</a><a id="2168" href="Categories.Morphism.html#1586" class="Field">Iso.isoˡ</a> <a id="2177" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="2179" href="Categories.Morphism.html#2040" class="Field">_≅_.iso</a> <a id="2187" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="2189" href="Categories.Diagram.Pullback.html#2216" class="Function">up-to-iso</a> <a id="2199" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="2201" href="Categories.Category.Extensive.html#1380" class="Bound">pb₁</a> <a id="2205" class="Symbol">(</a><a id="2206" href="Categories.Category.Extensive.html#847" class="Field">pullback₁</a> <a id="2216" href="Categories.Category.Extensive.html#1364" class="Bound">f</a><a id="2217" class="Symbol">))</a> <a id="2220" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
         <a id="2232" href="Categories.Object.Coproduct.html#1233" class="Function Operator">[</a> <a id="2234" href="Categories.Category.Extensive.html#2048" class="Bound">g</a> <a id="2236" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2238" href="Categories.Category.Extensive.html#1600" class="Function">pb₁-to-can</a> <a id="2249" href="Categories.Object.Coproduct.html#1233" class="Function Operator">,</a> <a id="2251" href="Categories.Category.Extensive.html#2051" class="Bound">h</a> <a id="2253" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2255" href="Categories.Category.Extensive.html#1685" class="Function">pb₂-to-can</a> <a id="2266" href="Categories.Object.Coproduct.html#1233" class="Function Operator">]</a> <a id="2268" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2270" class="Symbol">(</a><a id="2271" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="2274" href="Categories.Category.Extensive.html#1380" class="Bound">pb₁</a> <a id="2278" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2280" href="Categories.Category.Extensive.html#1600" class="Function">pb₁-to-can</a><a id="2290" class="Symbol">)</a> <a id="2292" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2294" href="Categories.Category.Extensive.html#1721" class="Function">can-to-pb₁</a>    <a id="2308" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2311" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="2319" href="Categories.Category.Extensive.html#1620" class="Function">p₁-≈₁</a> <a id="2325" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="2333" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
         <a id="2344" href="Categories.Object.Coproduct.html#1233" class="Function Operator">[</a> <a id="2346" href="Categories.Category.Extensive.html#2048" class="Bound">g</a> <a id="2348" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2350" href="Categories.Category.Extensive.html#1600" class="Function">pb₁-to-can</a> <a id="2361" href="Categories.Object.Coproduct.html#1233" class="Function Operator">,</a> <a id="2363" href="Categories.Category.Extensive.html#2051" class="Bound">h</a> <a id="2365" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2367" href="Categories.Category.Extensive.html#1685" class="Function">pb₂-to-can</a> <a id="2378" href="Categories.Object.Coproduct.html#1233" class="Function Operator">]</a> <a id="2380" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2382" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="2385" class="Symbol">(</a><a id="2386" href="Categories.Category.Extensive.html#847" class="Field">pullback₁</a> <a id="2396" href="Categories.Category.Extensive.html#1364" class="Bound">f</a><a id="2397" class="Symbol">)</a> <a id="2399" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2401" href="Categories.Category.Extensive.html#1721" class="Function">can-to-pb₁</a>         <a id="2420" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2423" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="2429" href="Categories.Object.Coproduct.html#1270" class="Function">inject₁</a> <a id="2437" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
         <a id="2448" class="Symbol">(</a><a id="2449" href="Categories.Category.Extensive.html#2048" class="Bound">g</a> <a id="2451" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2453" href="Categories.Category.Extensive.html#1600" class="Function">pb₁-to-can</a><a id="2463" class="Symbol">)</a> <a id="2465" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2467" href="Categories.Category.Extensive.html#1721" class="Function">can-to-pb₁</a>                                               <a id="2524" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2527" href="Categories.Morphism.Reasoning.Core.html#6665" class="Function">cancelʳ</a> <a id="2535" class="Symbol">(</a><a id="2536" href="Categories.Morphism.html#1586" class="Field">Iso.isoˡ</a> <a id="2545" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="2547" href="Categories.Morphism.html#2040" class="Field">_≅_.iso</a> <a id="2555" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="2557" href="Categories.Diagram.Pullback.html#2216" class="Function">up-to-iso</a> <a id="2567" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="2569" href="Categories.Category.Extensive.html#1380" class="Bound">pb₁</a> <a id="2573" class="Symbol">(</a><a id="2574" href="Categories.Category.Extensive.html#847" class="Field">pullback₁</a> <a id="2584" href="Categories.Category.Extensive.html#1364" class="Bound">f</a><a id="2585" class="Symbol">))</a> <a id="2588" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
         <a id="2600" href="Categories.Category.Extensive.html#2048" class="Bound">g</a>                                                                           <a id="2676" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>

      <a id="2685" href="Categories.Object.Coproduct.html#1303" class="Field">IsCoproduct.inject₂</a> <a id="2705" href="Categories.Category.Extensive.html#1844" class="Function">pullback-of-cp-is-cp&#39;</a> <a id="2727" class="Symbol">{_}{</a><a id="2731" href="Categories.Category.Extensive.html#2731" class="Bound">g</a><a id="2732" class="Symbol">}{</a><a id="2734" href="Categories.Category.Extensive.html#2734" class="Bound">h</a><a id="2735" class="Symbol">}</a> <a id="2737" class="Symbol">=</a> <a id="2739" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
         <a id="2754" href="Categories.Object.Coproduct.html#1233" class="Function Operator">[</a> <a id="2756" href="Categories.Category.Extensive.html#2731" class="Bound">g</a> <a id="2758" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2760" href="Categories.Category.Extensive.html#1600" class="Function">pb₁-to-can</a> <a id="2771" href="Categories.Object.Coproduct.html#1233" class="Function Operator">,</a> <a id="2773" href="Categories.Category.Extensive.html#2734" class="Bound">h</a> <a id="2775" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2777" href="Categories.Category.Extensive.html#1685" class="Function">pb₂-to-can</a> <a id="2788" href="Categories.Object.Coproduct.html#1233" class="Function Operator">]</a> <a id="2790" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2792" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="2795" href="Categories.Category.Extensive.html#1404" class="Bound">pb₂</a>                               <a id="2829" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="2833" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="2841" href="Categories.Morphism.Reasoning.Core.html#6665" class="Function">cancelʳ</a> <a id="2849" class="Symbol">(</a><a id="2850" href="Categories.Morphism.html#1586" class="Field">Iso.isoˡ</a> <a id="2859" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="2861" href="Categories.Morphism.html#2040" class="Field">_≅_.iso</a> <a id="2869" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="2871" href="Categories.Diagram.Pullback.html#2216" class="Function">up-to-iso</a> <a id="2881" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="2883" href="Categories.Category.Extensive.html#1404" class="Bound">pb₂</a> <a id="2887" class="Symbol">(</a><a id="2888" href="Categories.Category.Extensive.html#911" class="Field">pullback₂</a> <a id="2898" href="Categories.Category.Extensive.html#1364" class="Bound">f</a><a id="2899" class="Symbol">))</a> <a id="2902" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
         <a id="2914" href="Categories.Object.Coproduct.html#1233" class="Function Operator">[</a> <a id="2916" href="Categories.Category.Extensive.html#2731" class="Bound">g</a> <a id="2918" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2920" href="Categories.Category.Extensive.html#1600" class="Function">pb₁-to-can</a> <a id="2931" href="Categories.Object.Coproduct.html#1233" class="Function Operator">,</a> <a id="2933" href="Categories.Category.Extensive.html#2734" class="Bound">h</a> <a id="2935" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2937" href="Categories.Category.Extensive.html#1685" class="Function">pb₂-to-can</a> <a id="2948" href="Categories.Object.Coproduct.html#1233" class="Function Operator">]</a> <a id="2950" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2952" class="Symbol">(</a><a id="2953" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="2956" href="Categories.Category.Extensive.html#1404" class="Bound">pb₂</a> <a id="2960" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2962" href="Categories.Category.Extensive.html#1685" class="Function">pb₂-to-can</a><a id="2972" class="Symbol">)</a> <a id="2974" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2976" href="Categories.Category.Extensive.html#1783" class="Function">can-to-pb₂</a>    <a id="2990" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2993" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="3001" href="Categories.Category.Extensive.html#1705" class="Function">p₁-≈₂</a> <a id="3007" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="3015" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
         <a id="3026" href="Categories.Object.Coproduct.html#1233" class="Function Operator">[</a> <a id="3028" href="Categories.Category.Extensive.html#2731" class="Bound">g</a> <a id="3030" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3032" href="Categories.Category.Extensive.html#1600" class="Function">pb₁-to-can</a> <a id="3043" href="Categories.Object.Coproduct.html#1233" class="Function Operator">,</a> <a id="3045" href="Categories.Category.Extensive.html#2734" class="Bound">h</a> <a id="3047" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3049" href="Categories.Category.Extensive.html#1685" class="Function">pb₂-to-can</a> <a id="3060" href="Categories.Object.Coproduct.html#1233" class="Function Operator">]</a> <a id="3062" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3064" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="3067" class="Symbol">(</a><a id="3068" href="Categories.Category.Extensive.html#911" class="Field">pullback₂</a> <a id="3078" href="Categories.Category.Extensive.html#1364" class="Bound">f</a><a id="3079" class="Symbol">)</a> <a id="3081" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3083" href="Categories.Category.Extensive.html#1783" class="Function">can-to-pb₂</a>         <a id="3102" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3105" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="3111" href="Categories.Object.Coproduct.html#1303" class="Function">inject₂</a> <a id="3119" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
         <a id="3130" class="Symbol">(</a><a id="3131" href="Categories.Category.Extensive.html#2734" class="Bound">h</a> <a id="3133" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3135" href="Categories.Category.Extensive.html#1685" class="Function">pb₂-to-can</a><a id="3145" class="Symbol">)</a> <a id="3147" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3149" href="Categories.Category.Extensive.html#1783" class="Function">can-to-pb₂</a>                                               <a id="3206" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3209" href="Categories.Morphism.Reasoning.Core.html#6665" class="Function">cancelʳ</a> <a id="3217" class="Symbol">(</a><a id="3218" href="Categories.Morphism.html#1586" class="Field">Iso.isoˡ</a> <a id="3227" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="3229" href="Categories.Morphism.html#2040" class="Field">_≅_.iso</a> <a id="3237" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="3239" href="Categories.Diagram.Pullback.html#2216" class="Function">up-to-iso</a> <a id="3249" href="Categories.Category.Extensive.html#626" class="Bound">𝒞</a> <a id="3251" href="Categories.Category.Extensive.html#1404" class="Bound">pb₂</a> <a id="3255" class="Symbol">(</a><a id="3256" href="Categories.Category.Extensive.html#911" class="Field">pullback₂</a> <a id="3266" href="Categories.Category.Extensive.html#1364" class="Bound">f</a><a id="3267" class="Symbol">))</a> <a id="3270" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
         <a id="3282" href="Categories.Category.Extensive.html#2734" class="Bound">h</a>                                                                           <a id="3358" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>

      <a id="3375" href="Categories.Object.Coproduct.html#1336" class="Field">IsCoproduct.unique</a> <a id="3394" href="Categories.Category.Extensive.html#1844" class="Function">pullback-of-cp-is-cp&#39;</a>   <a id="3418" class="Symbol">{_}{</a><a id="3422" href="Categories.Category.Extensive.html#3422" class="Bound">u</a><a id="3423" class="Symbol">}{</a><a id="3425" href="Categories.Category.Extensive.html#3425" class="Bound">g</a><a id="3426" class="Symbol">}{</a><a id="3428" href="Categories.Category.Extensive.html#3428" class="Bound">h</a><a id="3429" class="Symbol">}</a>  <a id="3432" href="Categories.Category.Extensive.html#3432" class="Bound">u∘p₁pb₁≈g</a> <a id="3442" href="Categories.Category.Extensive.html#3442" class="Bound">u∘p₁pb₂≈h</a>  <a id="3453" class="Symbol">=</a> <a id="3455" href="Categories.Object.Coproduct.html#1336" class="Function">unique</a>
        <a id="3470" class="Symbol">(</a><a id="3471" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
                     <a id="3498" href="Categories.Category.Extensive.html#3422" class="Bound">u</a> <a id="3500" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3502" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="3505" class="Symbol">(</a><a id="3506" href="Categories.Category.Extensive.html#847" class="Field">pullback₁</a> <a id="3516" href="Categories.Category.Extensive.html#1364" class="Bound">f</a><a id="3517" class="Symbol">)</a>            <a id="3530" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="3534" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="3540" href="Categories.Category.Extensive.html#1620" class="Function">p₁-≈₁</a> <a id="3546" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
                     <a id="3570" class="Symbol">(</a><a id="3571" href="Categories.Category.Extensive.html#3422" class="Bound">u</a> <a id="3573" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3575" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="3578" href="Categories.Category.Extensive.html#1380" class="Bound">pb₁</a><a id="3581" class="Symbol">)</a> <a id="3583" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3585" href="Categories.Category.Extensive.html#1600" class="Function">pb₁-to-can</a>       <a id="3602" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3605" href="Categories.Category.Extensive.html#3432" class="Bound">u∘p₁pb₁≈g</a> <a id="3615" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="3623" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
                     <a id="3646" href="Categories.Category.Extensive.html#3425" class="Bound">g</a> <a id="3648" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3650" href="Categories.Category.Extensive.html#1600" class="Function">pb₁-to-can</a>                  <a id="3678" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a><a id="3679" class="Symbol">)</a>
        <a id="3689" class="Symbol">(</a><a id="3690" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
                     <a id="3717" href="Categories.Category.Extensive.html#3422" class="Bound">u</a> <a id="3719" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3721" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="3724" class="Symbol">(</a><a id="3725" href="Categories.Category.Extensive.html#911" class="Field">pullback₂</a> <a id="3735" href="Categories.Category.Extensive.html#1364" class="Bound">f</a><a id="3736" class="Symbol">)</a>           <a id="3748" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="3752" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="3758" href="Categories.Category.Extensive.html#1705" class="Function">p₁-≈₂</a> <a id="3764" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
                     <a id="3788" class="Symbol">(</a><a id="3789" href="Categories.Category.Extensive.html#3422" class="Bound">u</a> <a id="3791" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3793" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="3796" href="Categories.Category.Extensive.html#1404" class="Bound">pb₂</a><a id="3799" class="Symbol">)</a> <a id="3801" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3803" href="Categories.Category.Extensive.html#1685" class="Function">pb₂-to-can</a>       <a id="3820" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3823" href="Categories.Category.Extensive.html#3442" class="Bound">u∘p₁pb₂≈h</a> <a id="3833" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="3841" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
                     <a id="3864" href="Categories.Category.Extensive.html#3428" class="Bound">h</a> <a id="3866" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3868" href="Categories.Category.Extensive.html#1685" class="Function">pb₂-to-can</a>                  <a id="3896" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a><a id="3897" class="Symbol">)</a>

</pre></body></html>