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bsc-leon-vatthauer/agda/bsc-thesis/Categories.Diagram.Pullback.html
Leon Vatthauer 2bf4655eae
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2024-02-09 17:53:52 +01:00

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<!DOCTYPE HTML>
<html><head><meta charset="utf-8"><title>Categories.Diagram.Pullback</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.Core.html" class="Module">Categories.Category.Core</a> <a id="73" class="Keyword">using</a> <a id="79" class="Symbol">(</a><a id="80" href="Categories.Category.Core.html#442" class="Record">Category</a><a id="88" class="Symbol">)</a>
<a id="91" class="Keyword">module</a> <a id="98" href="Categories.Diagram.Pullback.html" class="Module">Categories.Diagram.Pullback</a> <a id="126" class="Symbol">{</a><a id="127" href="Categories.Diagram.Pullback.html#127" class="Bound">o</a> <a id="129" href="Categories.Diagram.Pullback.html#129" class="Bound">ℓ</a> <a id="131" href="Categories.Diagram.Pullback.html#131" class="Bound">e</a><a id="132" class="Symbol">}</a> <a id="134" class="Symbol">(</a><a id="135" href="Categories.Diagram.Pullback.html#135" class="Bound">C</a> <a id="137" class="Symbol">:</a> <a id="139" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="148" href="Categories.Diagram.Pullback.html#127" class="Bound">o</a> <a id="150" href="Categories.Diagram.Pullback.html#129" class="Bound">ℓ</a> <a id="152" href="Categories.Diagram.Pullback.html#131" class="Bound">e</a><a id="153" class="Symbol">)</a> <a id="155" class="Keyword">where</a>
<a id="162" class="Keyword">open</a> <a id="167" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="176" href="Categories.Diagram.Pullback.html#135" class="Bound">C</a>
<a id="178" class="Keyword">open</a> <a id="183" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="196" class="Keyword">open</a> <a id="201" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="208" class="Keyword">open</a> <a id="213" class="Keyword">import</a> <a id="220" href="Level.html" class="Module">Level</a>
<a id="226" class="Keyword">open</a> <a id="231" class="Keyword">import</a> <a id="238" href="Data.Product.html" class="Module">Data.Product</a> <a id="251" class="Keyword">using</a> <a id="257" class="Symbol">(</a><a id="258" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="261" class="Symbol">;</a> <a id="263" href="Data.Product.Base.html#852" class="Function">∃</a><a id="264" class="Symbol">)</a>
<a id="266" class="Keyword">open</a> <a id="271" class="Keyword">import</a> <a id="278" href="Function.html" class="Module">Function</a> <a id="287" class="Keyword">using</a> <a id="293" class="Symbol">(</a><a id="294" href="Function.Base.html#1638" class="Function">flip</a><a id="298" class="Symbol">;</a> <a id="300" href="Function.Base.html#1974" class="Function Operator">_$_</a><a id="303" class="Symbol">)</a> <a id="305" class="Keyword">renaming</a> <a id="314" class="Symbol">(</a><a id="315" href="Function.Base.html#1115" class="Function Operator">_∘_</a> <a id="319" class="Symbol">to</a> <a id="322" class="Function Operator">_●_</a><a id="325" class="Symbol">)</a>
<a id="327" class="Keyword">open</a> <a id="332" class="Keyword">import</a> <a id="339" href="Categories.Morphism.html" class="Module">Categories.Morphism</a> <a id="359" href="Categories.Diagram.Pullback.html#135" class="Bound">C</a>
<a id="361" class="Keyword">open</a> <a id="366" class="Keyword">import</a> <a id="373" href="Categories.Object.Product.html" class="Module">Categories.Object.Product</a> <a id="399" href="Categories.Diagram.Pullback.html#135" class="Bound">C</a> <a id="401" class="Keyword">hiding</a> <a id="408" class="Symbol">(</a><a id="409" href="Categories.Object.Product.Core.html#2581" class="Function">up-to-iso</a><a id="418" class="Symbol">;</a> <a id="420" href="Categories.Object.Product.Core.html#2078" class="Function">repack</a><a id="426" class="Symbol">;</a> <a id="428" href="Categories.Object.Product.Core.html#2168" class="Function">repack∘</a><a id="435" class="Symbol">;</a> <a id="437" href="Categories.Object.Product.Core.html#2451" class="Function">repack-cancel</a><a id="450" class="Symbol">)</a>
<a id="452" class="Keyword">open</a> <a id="457" class="Keyword">import</a> <a id="464" href="Categories.Diagram.Equalizer.html" class="Module">Categories.Diagram.Equalizer</a> <a id="493" href="Categories.Diagram.Pullback.html#135" class="Bound">C</a> <a id="495" class="Keyword">hiding</a> <a id="502" class="Symbol">(</a><a id="503" href="Categories.Diagram.Equalizer.html#2254" class="Function">up-to-iso</a><a id="512" class="Symbol">)</a>
<a id="514" class="Keyword">open</a> <a id="519" class="Keyword">import</a> <a id="526" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="556" href="Categories.Diagram.Pullback.html#135" class="Bound">C</a> as <a id="Square"></a><a id="561" href="Categories.Diagram.Pullback.html#561" class="Module">Square</a>
<a id="570" class="Keyword">renaming</a> <a id="579" class="Symbol">(</a><a id="580" href="Categories.Morphism.Reasoning.Core.html#4290" class="Function">glue</a> <a id="585" class="Symbol">to</a> <a id="588" class="Function">glue-square</a><a id="599" class="Symbol">)</a> <a id="601" class="Keyword">hiding</a> <a id="608" class="Symbol">(</a><a id="609" href="Categories.Morphism.Reasoning.Core.html#1075" class="Function">id-unique</a><a id="618" class="Symbol">)</a>
<a id="621" class="Keyword">private</a>
<a id="631" class="Keyword">variable</a>
<a id="644" href="Categories.Diagram.Pullback.html#644" class="Generalizable">A</a> <a id="646" href="Categories.Diagram.Pullback.html#646" class="Generalizable">B</a> <a id="648" href="Categories.Diagram.Pullback.html#648" class="Generalizable">X</a> <a id="650" href="Categories.Diagram.Pullback.html#650" class="Generalizable">Y</a> <a id="652" href="Categories.Diagram.Pullback.html#652" class="Generalizable">Z</a> <a id="654" class="Symbol">:</a> <a id="656" href="Categories.Category.Core.html#559" class="Field">Obj</a>
<a id="664" href="Categories.Diagram.Pullback.html#664" class="Generalizable">f</a> <a id="666" href="Categories.Diagram.Pullback.html#666" class="Generalizable">g</a> <a id="668" href="Categories.Diagram.Pullback.html#668" class="Generalizable">h</a> <a id="670" href="Categories.Diagram.Pullback.html#670" class="Generalizable">h₁</a> <a id="673" href="Categories.Diagram.Pullback.html#673" class="Generalizable">h₂</a> <a id="676" href="Categories.Diagram.Pullback.html#676" class="Generalizable">i</a> <a id="678" href="Categories.Diagram.Pullback.html#678" class="Generalizable">i₁</a> <a id="681" href="Categories.Diagram.Pullback.html#681" class="Generalizable">i₂</a> <a id="684" href="Categories.Diagram.Pullback.html#684" class="Generalizable">j</a> <a id="686" href="Categories.Diagram.Pullback.html#686" class="Generalizable">k</a> <a id="688" class="Symbol">:</a> <a id="690" href="Categories.Diagram.Pullback.html#644" class="Generalizable">A</a> <a id="692" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="694" href="Categories.Diagram.Pullback.html#646" class="Generalizable">B</a>
<a id="697" class="Comment">-- Proof that a given square is a pullback</a>
<a id="740" class="Keyword">record</a> <a id="IsPullback"></a><a id="747" href="Categories.Diagram.Pullback.html#747" class="Record">IsPullback</a> <a id="758" class="Symbol">{</a><a id="759" href="Categories.Diagram.Pullback.html#759" class="Bound">P</a> <a id="761" class="Symbol">:</a> <a id="763" href="Categories.Category.Core.html#559" class="Field">Obj</a><a id="766" class="Symbol">}</a> <a id="768" class="Symbol">(</a><a id="769" href="Categories.Diagram.Pullback.html#769" class="Bound">p₁</a> <a id="772" class="Symbol">:</a> <a id="774" href="Categories.Diagram.Pullback.html#759" class="Bound">P</a> <a id="776" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="778" href="Categories.Diagram.Pullback.html#648" class="Generalizable">X</a><a id="779" class="Symbol">)</a> <a id="781" class="Symbol">(</a><a id="782" href="Categories.Diagram.Pullback.html#782" class="Bound">p₂</a> <a id="785" class="Symbol">:</a> <a id="787" href="Categories.Diagram.Pullback.html#759" class="Bound">P</a> <a id="789" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="791" href="Categories.Diagram.Pullback.html#650" class="Generalizable">Y</a><a id="792" class="Symbol">)</a> <a id="794" class="Symbol">(</a><a id="795" href="Categories.Diagram.Pullback.html#795" class="Bound">f</a> <a id="797" class="Symbol">:</a> <a id="799" href="Categories.Diagram.Pullback.html#648" class="Generalizable">X</a> <a id="801" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="803" href="Categories.Diagram.Pullback.html#652" class="Generalizable">Z</a><a id="804" class="Symbol">)</a> <a id="806" class="Symbol">(</a><a id="807" href="Categories.Diagram.Pullback.html#807" class="Bound">g</a> <a id="809" class="Symbol">:</a> <a id="811" href="Categories.Diagram.Pullback.html#650" class="Generalizable">Y</a> <a id="813" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="815" href="Categories.Diagram.Pullback.html#652" class="Generalizable">Z</a><a id="816" class="Symbol">)</a> <a id="818" class="Symbol">:</a> <a id="820" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="824" class="Symbol">(</a><a id="825" href="Categories.Diagram.Pullback.html#127" class="Bound">o</a> <a id="827" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="829" href="Categories.Diagram.Pullback.html#129" class="Bound">ℓ</a> <a id="831" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="833" href="Categories.Diagram.Pullback.html#131" class="Bound">e</a><a id="834" class="Symbol">)</a> <a id="836" class="Keyword">where</a>
<a id="844" class="Keyword">field</a>
<a id="IsPullback.commute"></a><a id="854" href="Categories.Diagram.Pullback.html#854" class="Field">commute</a> <a id="864" class="Symbol">:</a> <a id="866" href="Categories.Diagram.Pullback.html#795" class="Bound">f</a> <a id="868" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="870" href="Categories.Diagram.Pullback.html#769" class="Bound">p₁</a> <a id="873" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="875" href="Categories.Diagram.Pullback.html#807" class="Bound">g</a> <a id="877" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="879" href="Categories.Diagram.Pullback.html#782" class="Bound">p₂</a>
<a id="IsPullback.universal"></a><a id="886" href="Categories.Diagram.Pullback.html#886" class="Field">universal</a> <a id="896" class="Symbol">:</a> <a id="898" class="Symbol">∀</a> <a id="900" class="Symbol">{</a><a id="901" href="Categories.Diagram.Pullback.html#901" class="Bound">h₁</a> <a id="904" class="Symbol">:</a> <a id="906" href="Categories.Diagram.Pullback.html#644" class="Generalizable">A</a> <a id="908" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="910" href="Categories.Diagram.Pullback.html#778" class="Bound">X</a><a id="911" class="Symbol">}</a> <a id="913" class="Symbol">{</a><a id="914" href="Categories.Diagram.Pullback.html#914" class="Bound">h₂</a> <a id="917" class="Symbol">:</a> <a id="919" href="Categories.Diagram.Pullback.html#644" class="Generalizable">A</a> <a id="921" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="923" href="Categories.Diagram.Pullback.html#791" class="Bound">Y</a><a id="924" class="Symbol">}</a> <a id="926" class="Symbol">→</a> <a id="928" href="Categories.Diagram.Pullback.html#795" class="Bound">f</a> <a id="930" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="932" href="Categories.Diagram.Pullback.html#901" class="Bound">h₁</a> <a id="935" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="937" href="Categories.Diagram.Pullback.html#807" class="Bound">g</a> <a id="939" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="941" href="Categories.Diagram.Pullback.html#914" class="Bound">h₂</a> <a id="944" class="Symbol">→</a> <a id="946" href="Categories.Diagram.Pullback.html#644" class="Generalizable">A</a> <a id="948" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="950" href="Categories.Diagram.Pullback.html#759" class="Bound">P</a>
<a id="IsPullback.unique"></a><a id="956" href="Categories.Diagram.Pullback.html#956" class="Field">unique</a> <a id="966" class="Symbol">:</a> <a id="968" class="Symbol">∀</a> <a id="970" class="Symbol">{</a><a id="971" href="Categories.Diagram.Pullback.html#971" class="Bound">eq</a> <a id="974" class="Symbol">:</a> <a id="976" href="Categories.Diagram.Pullback.html#795" class="Bound">f</a> <a id="978" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="980" href="Categories.Diagram.Pullback.html#670" class="Generalizable">h₁</a> <a id="983" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="985" href="Categories.Diagram.Pullback.html#807" class="Bound">g</a> <a id="987" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="989" href="Categories.Diagram.Pullback.html#673" class="Generalizable">h₂</a><a id="991" class="Symbol">}</a> <a id="993" class="Symbol">→</a>
<a id="1013" href="Categories.Diagram.Pullback.html#769" class="Bound">p₁</a> <a id="1016" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1018" href="Categories.Diagram.Pullback.html#676" class="Generalizable">i</a> <a id="1020" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1022" href="Categories.Diagram.Pullback.html#670" class="Generalizable">h₁</a> <a id="1025" class="Symbol">→</a> <a id="1027" href="Categories.Diagram.Pullback.html#782" class="Bound">p₂</a> <a id="1030" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1032" href="Categories.Diagram.Pullback.html#676" class="Generalizable">i</a> <a id="1034" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1036" href="Categories.Diagram.Pullback.html#673" class="Generalizable">h₂</a> <a id="1039" class="Symbol">→</a>
<a id="1059" href="Categories.Diagram.Pullback.html#676" class="Generalizable">i</a> <a id="1061" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1063" href="Categories.Diagram.Pullback.html#886" class="Field">universal</a> <a id="1073" href="Categories.Diagram.Pullback.html#971" class="Bound">eq</a>
<a id="IsPullback.p₁∘universal≈h₁"></a><a id="1081" href="Categories.Diagram.Pullback.html#1081" class="Field">p₁∘universal≈h₁</a> <a id="1098" class="Symbol">:</a> <a id="1100" class="Symbol">∀</a> <a id="1102" class="Symbol">{</a><a id="1103" href="Categories.Diagram.Pullback.html#1103" class="Bound">eq</a> <a id="1106" class="Symbol">:</a> <a id="1108" href="Categories.Diagram.Pullback.html#795" class="Bound">f</a> <a id="1110" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1112" href="Categories.Diagram.Pullback.html#670" class="Generalizable">h₁</a> <a id="1115" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1117" href="Categories.Diagram.Pullback.html#807" class="Bound">g</a> <a id="1119" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1121" href="Categories.Diagram.Pullback.html#673" class="Generalizable">h₂</a><a id="1123" class="Symbol">}</a> <a id="1125" class="Symbol">→</a>
<a id="1152" href="Categories.Diagram.Pullback.html#769" class="Bound">p₁</a> <a id="1155" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1157" href="Categories.Diagram.Pullback.html#886" class="Field">universal</a> <a id="1167" href="Categories.Diagram.Pullback.html#1103" class="Bound">eq</a> <a id="1170" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1172" href="Categories.Diagram.Pullback.html#670" class="Generalizable">h₁</a>
<a id="IsPullback.p₂∘universal≈h₂"></a><a id="1179" href="Categories.Diagram.Pullback.html#1179" class="Field">p₂∘universal≈h₂</a> <a id="1196" class="Symbol">:</a> <a id="1198" class="Symbol">∀</a> <a id="1200" class="Symbol">{</a><a id="1201" href="Categories.Diagram.Pullback.html#1201" class="Bound">eq</a> <a id="1204" class="Symbol">:</a> <a id="1206" href="Categories.Diagram.Pullback.html#795" class="Bound">f</a> <a id="1208" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1210" href="Categories.Diagram.Pullback.html#670" class="Generalizable">h₁</a> <a id="1213" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1215" href="Categories.Diagram.Pullback.html#807" class="Bound">g</a> <a id="1217" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1219" href="Categories.Diagram.Pullback.html#673" class="Generalizable">h₂</a><a id="1221" class="Symbol">}</a> <a id="1223" class="Symbol">→</a>
<a id="1250" href="Categories.Diagram.Pullback.html#782" class="Bound">p₂</a> <a id="1253" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1255" href="Categories.Diagram.Pullback.html#886" class="Field">universal</a> <a id="1265" href="Categories.Diagram.Pullback.html#1201" class="Bound">eq</a> <a id="1268" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1270" href="Categories.Diagram.Pullback.html#673" class="Generalizable">h₂</a>
<a id="IsPullback.unique′"></a><a id="1276" href="Categories.Diagram.Pullback.html#1276" class="Function">unique′</a> <a id="1284" class="Symbol">:</a> <a id="1286" class="Symbol">(</a><a id="1287" href="Categories.Diagram.Pullback.html#1287" class="Bound">eq</a> <a id="1290" href="Categories.Diagram.Pullback.html#1290" class="Bound">eq′</a> <a id="1294" class="Symbol">:</a> <a id="1296" href="Categories.Diagram.Pullback.html#795" class="Bound">f</a> <a id="1298" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1300" href="Categories.Diagram.Pullback.html#670" class="Generalizable">h₁</a> <a id="1303" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1305" href="Categories.Diagram.Pullback.html#807" class="Bound">g</a> <a id="1307" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1309" href="Categories.Diagram.Pullback.html#673" class="Generalizable">h₂</a><a id="1311" class="Symbol">)</a> <a id="1313" class="Symbol">→</a> <a id="1315" href="Categories.Diagram.Pullback.html#886" class="Field">universal</a> <a id="1325" href="Categories.Diagram.Pullback.html#1287" class="Bound">eq</a> <a id="1328" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1330" href="Categories.Diagram.Pullback.html#886" class="Field">universal</a> <a id="1340" href="Categories.Diagram.Pullback.html#1290" class="Bound">eq′</a>
<a id="1346" href="Categories.Diagram.Pullback.html#1276" class="Function">unique′</a> <a id="1354" href="Categories.Diagram.Pullback.html#1354" class="Bound">eq</a> <a id="1357" href="Categories.Diagram.Pullback.html#1357" class="Bound">eq′</a> <a id="1361" class="Symbol">=</a> <a id="1363" href="Categories.Diagram.Pullback.html#956" class="Field">unique</a> <a id="1370" href="Categories.Diagram.Pullback.html#1081" class="Field">p₁∘universal≈h₁</a> <a id="1386" href="Categories.Diagram.Pullback.html#1179" class="Field">p₂∘universal≈h₂</a>
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<a id="4743" class="Symbol">(</a><a id="4744" href="Categories.Diagram.Pullback.html#4524" class="Bound">f</a> <a id="4746" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4748" href="Categories.Diagram.Pullback.html#4540" class="Bound">h</a><a id="4749" class="Symbol">)</a> <a id="4751" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4753" href="Categories.Diagram.Pullback.html#4705" class="Bound">h₁</a> <a id="4761" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4764" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="4770" href="Categories.Diagram.Pullback.html#4712" class="Bound">eq</a> <a id="4773" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4781" href="Categories.Diagram.Pullback.html#4524" class="Bound">f</a> <a id="4783" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4785" href="Categories.Diagram.Pullback.html#2112" class="Function">p.p₁</a> <a id="4790" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4792" href="Categories.Diagram.Pullback.html#4708" class="Bound">h₂</a> <a id="4799" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4802" href="Categories.Morphism.Reasoning.Core.html#3587" class="Function">extendʳ</a> <a id="4810" href="Categories.Diagram.Pullback.html#854" class="Function">p.commute</a> <a id="4820" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4828" href="Categories.Diagram.Pullback.html#4532" class="Bound">g</a> <a id="4830" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4832" href="Categories.Diagram.Pullback.html#2128" class="Function">p.p₂</a> <a id="4837" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4839" href="Categories.Diagram.Pullback.html#4708" class="Bound">h₂</a> <a id="4846" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="4852" class="Symbol">;</a> <a id="4854" href="Categories.Diagram.Pullback.html#956" class="Field">unique</a> <a id="4870" class="Symbol">=</a> <a id="4872" class="Symbol">λ</a> <a id="4874" class="Symbol">{</a><a id="4875" href="Categories.Diagram.Pullback.html#4875" class="Bound">_</a> <a id="4877" href="Categories.Diagram.Pullback.html#4877" class="Bound">h₁</a> <a id="4880" href="Categories.Diagram.Pullback.html#4880" class="Bound">h₂</a> <a id="4883" href="Categories.Diagram.Pullback.html#4883" class="Bound">i</a><a id="4884" class="Symbol">}</a> <a id="4886" href="Categories.Diagram.Pullback.html#4886" class="Bound">eq</a> <a id="4889" href="Categories.Diagram.Pullback.html#4889" class="Bound">eq′</a> <a id="4893" class="Symbol">→</a> <a id="4895" href="Categories.Diagram.Pullback.html#956" class="Function">q.unique</a> <a id="4904" href="Categories.Diagram.Pullback.html#4886" class="Bound">eq</a> <a id="4907" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="4909" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="4919" href="Categories.Diagram.Pullback.html#2128" class="Field">q.p₂</a> <a id="4924" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4926" href="Categories.Diagram.Pullback.html#4883" class="Bound">i</a> <a id="4939" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4942" href="Categories.Morphism.Reasoning.Core.html#2485" class="Function">pushˡ</a> <a id="4948" class="Symbol">(</a><a id="4949" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="4951" href="Categories.Diagram.Pullback.html#1179" class="Function">p.p₂∘universal≈h₂</a><a id="4968" class="Symbol">)</a> <a id="4970" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4976" href="Categories.Diagram.Pullback.html#2128" class="Function">p.p₂</a> <a id="4981" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4983" href="Categories.Diagram.Pullback.html#5349" class="Function">p₂′</a> <a id="4987" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4989" href="Categories.Diagram.Pullback.html#4883" class="Bound">i</a> <a id="4996" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4999" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="5007" href="Categories.Diagram.Pullback.html#4889" class="Bound">eq′</a> <a id="5011" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5017" href="Categories.Diagram.Pullback.html#2128" class="Function">p.p₂</a> <a id="5022" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5024" href="Categories.Diagram.Pullback.html#4880" class="Bound">h₂</a> <a id="5037" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="5043" class="Symbol">;</a> <a id="5045" href="Categories.Diagram.Pullback.html#1081" class="Field">p₁∘universal≈h₁</a> <a id="5061" class="Symbol">=</a> <a id="5063" href="Categories.Diagram.Pullback.html#1081" class="Function">q.p₁∘universal≈h₁</a>
<a id="5085" class="Symbol">;</a> <a id="5087" href="Categories.Diagram.Pullback.html#1179" class="Field">p₂∘universal≈h₂</a> <a id="5103" class="Symbol">=</a> <a id="5105" class="Symbol">λ</a> <a id="5107" class="Symbol">{</a><a id="5108" href="Categories.Diagram.Pullback.html#5108" class="Bound">_</a> <a id="5110" href="Categories.Diagram.Pullback.html#5110" class="Bound">_</a> <a id="5112" href="Categories.Diagram.Pullback.html#5112" class="Bound">_</a> <a id="5114" href="Categories.Diagram.Pullback.html#5114" class="Bound">eq</a><a id="5116" class="Symbol">}</a> <a id="5118" class="Symbol">→</a>
<a id="5126" href="Categories.Diagram.Pullback.html#1698" class="Function">p.unique-diagram</a> <a id="5143" class="Symbol">((</a><a id="5145" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="5151" href="Categories.Diagram.Pullback.html#1081" class="Function">p.p₁∘universal≈h₁</a><a id="5168" class="Symbol">)</a> <a id="5170" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="5172" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="5178" href="Categories.Diagram.Pullback.html#1081" class="Function">q.p₁∘universal≈h₁</a> <a id="5196" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="5198" href="Categories.Diagram.Pullback.html#5114" class="Bound">eq</a><a id="5200" class="Symbol">)</a>
<a id="5225" class="Symbol">(</a><a id="5226" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="5232" href="Categories.Diagram.Pullback.html#1179" class="Function">p.p₂∘universal≈h₂</a> <a id="5250" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="5252" href="Categories.Diagram.Pullback.html#1179" class="Function">q.p₂∘universal≈h₂</a><a id="5269" class="Symbol">)</a>
<a id="5275" class="Symbol">}</a>
<a id="5279" class="Symbol">}</a>
<a id="5283" class="Keyword">where</a> <a id="5289" class="Keyword">module</a> <a id="5296" href="Categories.Diagram.Pullback.html#5296" class="Module">p</a> <a id="5298" class="Symbol">=</a> <a id="5300" href="Categories.Diagram.Pullback.html#2029" class="Module">Pullback</a> <a id="5309" href="Categories.Diagram.Pullback.html#4543" class="Bound">p</a>
<a id="5319" class="Keyword">module</a> <a id="5326" href="Categories.Diagram.Pullback.html#5326" class="Module">q</a> <a id="5328" class="Symbol">=</a> <a id="5330" href="Categories.Diagram.Pullback.html#2029" class="Module">Pullback</a> <a id="5339" href="Categories.Diagram.Pullback.html#4545" class="Bound">q</a>
<a id="5349" href="Categories.Diagram.Pullback.html#5349" class="Function">p₂′</a> <a id="5353" class="Symbol">=</a> <a id="5355" href="Categories.Diagram.Pullback.html#886" class="Function">p.universal</a> <a id="5367" class="Symbol">(</a><a id="5368" href="Categories.Category.Core.html#1004" class="Field">sym-assoc</a> <a id="5378" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="5380" href="Categories.Diagram.Pullback.html#854" class="Function">q.commute</a><a id="5389" class="Symbol">)</a> <a id="5391" class="Comment">-- used twice above</a>
<a id="Product×Equalizer⇒Pullback"></a><a id="5412" href="Categories.Diagram.Pullback.html#5412" class="Function">Product×Equalizer⇒Pullback</a> <a id="5439" class="Symbol">:</a>
<a id="5443" class="Symbol">(</a><a id="5444" href="Categories.Diagram.Pullback.html#5444" class="Bound">p</a> <a id="5446" class="Symbol">:</a> <a id="5448" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="5456" href="Categories.Diagram.Pullback.html#644" class="Generalizable">A</a> <a id="5458" href="Categories.Diagram.Pullback.html#646" class="Generalizable">B</a><a id="5459" class="Symbol">)</a> <a id="5461" class="Symbol">→</a> <a id="5463" href="Categories.Diagram.Equalizer.html#1948" class="Record">Equalizer</a> <a id="5473" class="Symbol">(</a><a id="5474" href="Categories.Diagram.Pullback.html#664" class="Generalizable">f</a> <a id="5476" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5478" href="Categories.Object.Product.Core.html#526" class="Field">Product.π₁</a> <a id="5489" href="Categories.Diagram.Pullback.html#5444" class="Bound">p</a><a id="5490" class="Symbol">)</a> <a id="5492" class="Symbol">(</a><a id="5493" href="Categories.Diagram.Pullback.html#666" class="Generalizable">g</a> <a id="5495" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5497" href="Categories.Object.Product.Core.html#546" class="Field">Product.π₂</a> <a id="5508" href="Categories.Diagram.Pullback.html#5444" class="Bound">p</a><a id="5509" class="Symbol">)</a> <a id="5511" class="Symbol">→</a>
<a id="5515" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="5524" href="Categories.Diagram.Pullback.html#664" class="Generalizable">f</a> <a id="5526" href="Categories.Diagram.Pullback.html#666" class="Generalizable">g</a>
<a id="5528" href="Categories.Diagram.Pullback.html#5412" class="Function">Product×Equalizer⇒Pullback</a> <a id="5555" class="Symbol">{</a><a id="5556" class="Argument">f</a> <a id="5558" class="Symbol">=</a> <a id="5560" href="Categories.Diagram.Pullback.html#5560" class="Bound">f</a><a id="5561" class="Symbol">}</a> <a id="5563" class="Symbol">{</a><a id="5564" class="Argument">g</a> <a id="5566" class="Symbol">=</a> <a id="5568" href="Categories.Diagram.Pullback.html#5568" class="Bound">g</a><a id="5569" class="Symbol">}</a> <a id="5571" href="Categories.Diagram.Pullback.html#5571" class="Bound">p</a> <a id="5573" href="Categories.Diagram.Pullback.html#5573" class="Bound">e</a> <a id="5575" class="Symbol">=</a> <a id="5577" class="Keyword">record</a>
<a id="5586" class="Symbol">{</a> <a id="5588" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="5604" class="Symbol">=</a> <a id="5606" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="5609" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5611" href="Categories.Diagram.Equalizer.html#2024" class="Field">arr</a>
<a id="5617" class="Symbol">;</a> <a id="5619" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="5635" class="Symbol">=</a> <a id="5637" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="5640" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5642" href="Categories.Diagram.Equalizer.html#2024" class="Field">arr</a>
<a id="5648" class="Symbol">;</a> <a id="5650" href="Categories.Diagram.Pullback.html#2144" class="Field">isPullback</a> <a id="5661" class="Symbol">=</a> <a id="5663" class="Keyword">record</a>
<a id="5674" class="Symbol">{</a> <a id="5676" href="Categories.Diagram.Pullback.html#854" class="Field">commute</a> <a id="5692" class="Symbol">=</a> <a id="5694" href="Categories.Category.Core.html#1004" class="Field">sym-assoc</a> <a id="5704" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="5706" href="Categories.Diagram.Equalizer.html#542" class="Function">equality</a> <a id="5715" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="5717" href="Categories.Category.Core.html#715" class="Field">assoc</a>
<a id="5727" class="Symbol">;</a> <a id="5729" href="Categories.Diagram.Pullback.html#886" class="Field">universal</a> <a id="5745" class="Symbol">=</a> <a id="5747" class="Symbol">λ</a> <a id="5749" class="Symbol">{</a><a id="5750" href="Categories.Diagram.Pullback.html#5750" class="Bound">_</a> <a id="5752" href="Categories.Diagram.Pullback.html#5752" class="Bound">h₁</a> <a id="5755" href="Categories.Diagram.Pullback.html#5755" class="Bound">h₂</a><a id="5757" class="Symbol">}</a> <a id="5759" href="Categories.Diagram.Pullback.html#5759" class="Bound">eq</a> <a id="5762" class="Symbol">→</a> <a id="5764" href="Categories.Diagram.Equalizer.html#575" class="Function">equalize</a> <a id="5773" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="5775" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="5787" class="Symbol">(</a><a id="5788" href="Categories.Diagram.Pullback.html#5560" class="Bound">f</a> <a id="5790" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5792" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a><a id="5794" class="Symbol">)</a> <a id="5796" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5798" href="Categories.Object.Product.Core.html#566" class="Function Operator">⟨</a> <a id="5800" href="Categories.Diagram.Pullback.html#5752" class="Bound">h₁</a> <a id="5803" href="Categories.Object.Product.Core.html#566" class="Function Operator">,</a> <a id="5805" href="Categories.Diagram.Pullback.html#5755" class="Bound">h₂</a> <a id="5808" href="Categories.Object.Product.Core.html#566" class="Function Operator">⟩</a> <a id="5810" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5813" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="5819" href="Categories.Object.Product.Core.html#603" class="Function">project₁</a> <a id="5828" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5836" href="Categories.Diagram.Pullback.html#5560" class="Bound">f</a> <a id="5838" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5840" href="Categories.Diagram.Pullback.html#5752" class="Bound">h₁</a> <a id="5858" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5861" href="Categories.Diagram.Pullback.html#5759" class="Bound">eq</a> <a id="5864" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5872" href="Categories.Diagram.Pullback.html#5568" class="Bound">g</a> <a id="5874" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5876" href="Categories.Diagram.Pullback.html#5755" class="Bound">h₂</a> <a id="5894" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="5898" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="5904" href="Categories.Object.Product.Core.html#637" class="Function">project₂</a> <a id="5913" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="5921" class="Symbol">(</a><a id="5922" href="Categories.Diagram.Pullback.html#5568" class="Bound">g</a> <a id="5924" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5926" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a><a id="5928" class="Symbol">)</a> <a id="5930" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5932" href="Categories.Object.Product.Core.html#566" class="Function Operator">⟨</a> <a id="5934" href="Categories.Diagram.Pullback.html#5752" class="Bound">h₁</a> <a id="5937" href="Categories.Object.Product.Core.html#566" class="Function Operator">,</a> <a id="5939" href="Categories.Diagram.Pullback.html#5755" class="Bound">h₂</a> <a id="5942" href="Categories.Object.Product.Core.html#566" class="Function Operator">⟩</a> <a id="5944" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="5950" class="Symbol">;</a> <a id="5952" href="Categories.Diagram.Pullback.html#956" class="Field">unique</a> <a id="5968" class="Symbol">=</a> <a id="5970" class="Symbol">λ</a> <a id="5972" href="Categories.Diagram.Pullback.html#5972" class="Bound">eq</a> <a id="5975" href="Categories.Diagram.Pullback.html#5975" class="Bound">eq′</a> <a id="5979" class="Symbol">→</a> <a id="5981" href="Categories.Diagram.Equalizer.html#691" class="Function">e.unique</a> <a id="5990" class="Symbol">(</a><a id="5991" href="Categories.Object.Product.Core.html#671" class="Function">p.unique</a> <a id="6000" class="Symbol">(</a><a id="6001" href="Categories.Category.Core.html#1004" class="Field">sym-assoc</a> <a id="6011" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="6013" href="Categories.Diagram.Pullback.html#5972" class="Bound">eq</a><a id="6015" class="Symbol">)</a> <a id="6017" class="Symbol">(</a><a id="6018" href="Categories.Category.Core.html#1004" class="Field">sym-assoc</a> <a id="6028" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="6030" href="Categories.Diagram.Pullback.html#5975" class="Bound">eq′</a><a id="6033" class="Symbol">))</a>
<a id="6040" class="Symbol">;</a> <a id="6042" href="Categories.Diagram.Pullback.html#1081" class="Field">p₁∘universal≈h₁</a> <a id="6058" class="Symbol">=</a> <a id="6060" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="6066" class="Symbol">(</a><a id="6067" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="6069" href="Categories.Diagram.Equalizer.html#628" class="Function">e.universal</a><a id="6080" class="Symbol">)</a> <a id="6082" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="6084" href="Categories.Object.Product.Core.html#603" class="Function">project₁</a>
<a id="6097" class="Symbol">;</a> <a id="6099" href="Categories.Diagram.Pullback.html#1179" class="Field">p₂∘universal≈h₂</a> <a id="6115" class="Symbol">=</a> <a id="6117" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="6123" class="Symbol">(</a><a id="6124" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="6126" href="Categories.Diagram.Equalizer.html#628" class="Function">e.universal</a><a id="6137" class="Symbol">)</a> <a id="6139" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="6141" href="Categories.Object.Product.Core.html#637" class="Function">project₂</a>
<a id="6154" class="Symbol">}</a>
<a id="6158" class="Symbol">}</a>
<a id="6162" class="Keyword">where</a> <a id="6168" class="Keyword">module</a> <a id="6175" href="Categories.Diagram.Pullback.html#6175" class="Module">p</a> <a id="6177" class="Symbol">=</a> <a id="6179" href="Categories.Object.Product.Core.html#435" class="Module">Product</a> <a id="6187" href="Categories.Diagram.Pullback.html#5571" class="Bound">p</a>
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<a id="7715" href="Categories.Diagram.Pullback.html#7715" class="Function">Pullback-resp-Mono</a> <a id="7734" class="Symbol">:</a> <a id="7736" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="7741" href="Categories.Diagram.Pullback.html#7236" class="Bound">g</a> <a id="7743" class="Symbol">→</a> <a id="7745" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="7750" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a>
<a id="7755" href="Categories.Diagram.Pullback.html#7715" class="Function">Pullback-resp-Mono</a> <a id="7774" href="Categories.Diagram.Pullback.html#7774" class="Bound">mg</a> <a id="7777" href="Categories.Diagram.Pullback.html#7777" class="Bound">h</a> <a id="7779" href="Categories.Diagram.Pullback.html#7779" class="Bound">i</a> <a id="7781" href="Categories.Diagram.Pullback.html#7781" class="Bound">eq</a> <a id="7784" class="Symbol">=</a> <a id="7786" href="Categories.Diagram.Pullback.html#1698" class="Function">unique-diagram</a> <a id="7801" href="Categories.Diagram.Pullback.html#7781" class="Bound">eq</a> <a id="7804" class="Symbol">(</a><a id="7805" href="Categories.Diagram.Pullback.html#7774" class="Bound">mg</a> <a id="7808" class="Symbol">_</a> <a id="7810" class="Symbol">_</a> <a id="7812" href="Categories.Diagram.Pullback.html#7827" class="Function">eq′</a><a id="7815" class="Symbol">)</a>
<a id="7821" class="Keyword">where</a> <a id="7827" href="Categories.Diagram.Pullback.html#7827" class="Function">eq′</a> <a id="7831" class="Symbol">:</a> <a id="7833" href="Categories.Diagram.Pullback.html#7236" class="Bound">g</a> <a id="7835" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="7837" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="7840" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="7842" href="Categories.Diagram.Pullback.html#7777" class="Bound">h</a> <a id="7844" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="7846" href="Categories.Diagram.Pullback.html#7236" class="Bound">g</a> <a id="7848" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="7850" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="7853" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="7855" href="Categories.Diagram.Pullback.html#7779" class="Bound">i</a>
<a id="7867" href="Categories.Diagram.Pullback.html#7827" class="Function">eq′</a> <a id="7871" class="Symbol">=</a> <a id="7873" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="7891" href="Categories.Diagram.Pullback.html#7236" class="Bound">g</a> <a id="7893" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="7895" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="7898" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="7900" href="Categories.Diagram.Pullback.html#7777" class="Bound">h</a> <a id="7902" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="7905" href="Categories.Morphism.Reasoning.Core.html#3587" class="Function">extendʳ</a> <a id="7913" class="Symbol">(</a><a id="7914" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="7918" href="Categories.Diagram.Pullback.html#854" class="Function">commute</a><a id="7925" class="Symbol">)</a> <a id="7927" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="7941" href="Categories.Diagram.Pullback.html#7234" class="Bound">f</a> <a id="7943" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="7945" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="7948" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="7950" href="Categories.Diagram.Pullback.html#7777" class="Bound">h</a> <a id="7952" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="7955" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="7963" href="Categories.Diagram.Pullback.html#7781" class="Bound">eq</a> <a id="7966" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="7980" href="Categories.Diagram.Pullback.html#7234" class="Bound">f</a> <a id="7982" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="7984" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="7987" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="7989" href="Categories.Diagram.Pullback.html#7779" class="Bound">i</a> <a id="7991" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="7994" href="Categories.Morphism.Reasoning.Core.html#3587" class="Function">extendʳ</a> <a id="8002" href="Categories.Diagram.Pullback.html#854" class="Function">commute</a> <a id="8010" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="8024" href="Categories.Diagram.Pullback.html#7236" class="Bound">g</a> <a id="8026" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8028" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="8031" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8033" href="Categories.Diagram.Pullback.html#7779" class="Bound">i</a> <a id="8035" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="8040" href="Categories.Diagram.Pullback.html#8040" class="Function">Pullback-resp-Iso</a> <a id="8058" class="Symbol">:</a> <a id="8060" href="Categories.Morphism.html#1528" class="Record">Iso</a> <a id="8064" href="Categories.Diagram.Pullback.html#7236" class="Bound">g</a> <a id="8066" href="Categories.Diagram.Pullback.html#668" class="Generalizable">h</a> <a id="8068" class="Symbol">→</a> <a id="8070" href="Data.Product.Base.html#852" class="Function">∃</a> <a id="8072" class="Symbol">λ</a> <a id="8074" href="Categories.Diagram.Pullback.html#8074" class="Bound">i</a> <a id="8076" class="Symbol">→</a> <a id="8078" href="Categories.Morphism.html#1528" class="Record">Iso</a> <a id="8082" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="8085" href="Categories.Diagram.Pullback.html#8074" class="Bound">i</a>
<a id="8089" href="Categories.Diagram.Pullback.html#8040" class="Function">Pullback-resp-Iso</a> <a id="8107" class="Symbol">{</a><a id="8108" class="Argument">h</a> <a id="8110" class="Symbol">=</a> <a id="8112" href="Categories.Diagram.Pullback.html#8112" class="Bound">h</a><a id="8113" class="Symbol">}</a> <a id="8115" href="Categories.Diagram.Pullback.html#8115" class="Bound">iso</a> <a id="8119" class="Symbol">=</a> <a id="8121" href="Categories.Diagram.Pullback.html#886" class="Function">universal</a> <a id="8131" href="Categories.Diagram.Pullback.html#8363" class="Function">eq</a>
<a id="8166" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="8168" class="Keyword">record</a>
<a id="8207" class="Symbol">{</a> <a id="8209" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="8214" class="Symbol">=</a> <a id="8216" href="Categories.Diagram.Pullback.html#1698" class="Function">unique-diagram</a> <a id="8231" href="Categories.Diagram.Pullback.html#8528" class="Function">eq₁</a> <a id="8235" href="Categories.Diagram.Pullback.html#8702" class="Function">eq₂</a>
<a id="8271" class="Symbol">;</a> <a id="8273" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="8278" class="Symbol">=</a> <a id="8280" href="Categories.Diagram.Pullback.html#1081" class="Function">p₁∘universal≈h₁</a>
<a id="8328" class="Symbol">}</a>
<a id="8334" class="Keyword">where</a> <a id="8340" class="Keyword">open</a> <a id="8345" href="Categories.Morphism.html#1528" class="Module">Iso</a> <a id="8349" href="Categories.Diagram.Pullback.html#8115" class="Bound">iso</a>
<a id="8363" href="Categories.Diagram.Pullback.html#8363" class="Function">eq</a> <a id="8366" class="Symbol">=</a> <a id="8368" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="8386" href="Categories.Diagram.Pullback.html#7234" class="Bound">f</a> <a id="8388" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8390" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="8409" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="8412" href="Categories.Morphism.Reasoning.Core.html#3063" class="Function">introˡ</a> <a id="8419" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="8424" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="8438" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="8441" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8443" href="Categories.Diagram.Pullback.html#7234" class="Bound">f</a> <a id="8445" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8447" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="8461" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="8464" href="Categories.Morphism.Reasoning.Core.html#2485" class="Function">pushˡ</a> <a id="8470" class="Symbol">(</a><a id="8471" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="8473" href="Categories.Morphism.html#1612" class="Field">isoʳ</a><a id="8477" class="Symbol">)</a> <a id="8479" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="8493" href="Categories.Diagram.Pullback.html#7236" class="Bound">g</a> <a id="8495" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8497" href="Categories.Diagram.Pullback.html#8112" class="Bound">h</a> <a id="8499" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8501" href="Categories.Diagram.Pullback.html#7234" class="Bound">f</a> <a id="8503" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8505" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="8516" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="8528" href="Categories.Diagram.Pullback.html#8528" class="Function">eq₁</a> <a id="8532" class="Symbol">=</a> <a id="8534" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="8552" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="8555" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8557" href="Categories.Diagram.Pullback.html#886" class="Function">universal</a> <a id="8567" href="Categories.Diagram.Pullback.html#8363" class="Function">eq</a> <a id="8570" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8572" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="8575" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="8578" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="8586" href="Categories.Diagram.Pullback.html#1081" class="Function">p₁∘universal≈h₁</a> <a id="8602" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="8616" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="8639" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="8643" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a> <a id="8653" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="8667" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="8670" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8672" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="8690" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="8702" href="Categories.Diagram.Pullback.html#8702" class="Function">eq₂</a> <a id="8706" class="Symbol">=</a> <a id="8708" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="8726" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="8729" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8731" href="Categories.Diagram.Pullback.html#886" class="Function">universal</a> <a id="8741" href="Categories.Diagram.Pullback.html#8363" class="Function">eq</a> <a id="8744" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8746" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="8749" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="8752" href="Categories.Morphism.Reasoning.Core.html#3587" class="Function">extendʳ</a> <a id="8760" href="Categories.Diagram.Pullback.html#1179" class="Function">p₂∘universal≈h₂</a> <a id="8776" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="8790" href="Categories.Diagram.Pullback.html#8112" class="Bound">h</a> <a id="8792" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8794" class="Symbol">(</a><a id="8795" href="Categories.Diagram.Pullback.html#7234" class="Bound">f</a> <a id="8797" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8799" href="Categories.Category.Core.html#630" class="Field">id</a><a id="8801" class="Symbol">)</a> <a id="8803" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8805" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="8813" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="8816" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="8821" href="Categories.Category.Core.html#2634" class="Function Operator">⟩∘⟨</a> <a id="8825" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a> <a id="8835" href="Categories.Category.Core.html#2634" class="Function Operator">⟩∘⟨</a> <a id="8839" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="8844" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="8858" href="Categories.Diagram.Pullback.html#8112" class="Bound">h</a> <a id="8860" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8862" href="Categories.Diagram.Pullback.html#7234" class="Bound">f</a> <a id="8864" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8866" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="8881" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="8884" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="8889" href="Categories.Category.Core.html#2634" class="Function Operator">⟩∘⟨</a> <a id="8893" href="Categories.Diagram.Pullback.html#854" class="Function">commute</a> <a id="8901" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="8915" href="Categories.Diagram.Pullback.html#8112" class="Bound">h</a> <a id="8917" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8919" href="Categories.Diagram.Pullback.html#7236" class="Bound">g</a> <a id="8921" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="8923" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="8938" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="8941" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="8949" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="8954" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="8968" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="8991" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="8995" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a> <a id="9005" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="9019" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="9022" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="9024" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="9042" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
</pre></body></html>
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