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<html><head><meta charset="utf-8"><title>Categories.Diagram.Equalizer</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="Comment">-- Equalizers in a Category C</a>
<a id="121" class="Keyword">module</a> <a id="128" href="Categories.Diagram.Equalizer.html" class="Module">Categories.Diagram.Equalizer</a> <a id="157" class="Symbol">{</a><a id="158" href="Categories.Diagram.Equalizer.html#158" class="Bound">o</a> <a id="160" href="Categories.Diagram.Equalizer.html#160" class="Bound">ℓ</a> <a id="162" href="Categories.Diagram.Equalizer.html#162" class="Bound">e</a><a id="163" class="Symbol">}</a> <a id="165" class="Symbol">(</a><a id="166" href="Categories.Diagram.Equalizer.html#166" class="Bound">C</a> <a id="168" class="Symbol">:</a> <a id="170" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="179" href="Categories.Diagram.Equalizer.html#158" class="Bound">o</a> <a id="181" href="Categories.Diagram.Equalizer.html#160" class="Bound">ℓ</a> <a id="183" href="Categories.Diagram.Equalizer.html#162" class="Bound">e</a><a id="184" class="Symbol">)</a> <a id="186" class="Keyword">where</a>
<a id="193" class="Keyword">open</a> <a id="198" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="207" href="Categories.Diagram.Equalizer.html#166" class="Bound">C</a>
<a id="209" class="Keyword">open</a> <a id="214" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="227" class="Keyword">open</a> <a id="232" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="239" class="Keyword">open</a> <a id="244" class="Keyword">import</a> <a id="251" href="Level.html" class="Module">Level</a>
<a id="257" class="Keyword">open</a> <a id="262" class="Keyword">import</a> <a id="269" href="Data.Product.html" class="Module">Data.Product</a> <a id="282" class="Symbol">as</a> <a id="285" class="Module">Σ</a>
<a id="287" class="Keyword">open</a> <a id="292" class="Keyword">import</a> <a id="299" href="Function.html" class="Module">Function</a> <a id="308" class="Keyword">using</a> <a id="314" class="Symbol">(</a><a id="315" href="Function.Base.html#1974" class="Function Operator">_$_</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.Morphism.html" class="Module">Categories.Morphism</a> <a id="352" href="Categories.Diagram.Equalizer.html#166" class="Bound">C</a>
<a id="354" class="Keyword">open</a> <a id="359" class="Keyword">import</a> <a id="366" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="396" href="Categories.Diagram.Equalizer.html#166" class="Bound">C</a>
<a id="399" class="Keyword">private</a>
<a id="409" class="Keyword">variable</a>
<a id="422" href="Categories.Diagram.Equalizer.html#422" class="Generalizable">A</a> <a id="424" href="Categories.Diagram.Equalizer.html#424" class="Generalizable">B</a> <a id="426" href="Categories.Diagram.Equalizer.html#426" class="Generalizable">X</a> <a id="428" class="Symbol">:</a> <a id="430" href="Categories.Category.Core.html#559" class="Field">Obj</a>
<a id="438" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a> <a id="440" href="Categories.Diagram.Equalizer.html#440" class="Generalizable">i</a> <a id="442" href="Categories.Diagram.Equalizer.html#442" class="Generalizable">j</a> <a id="444" href="Categories.Diagram.Equalizer.html#444" class="Generalizable">k</a> <a id="446" class="Symbol">:</a> <a id="448" href="Categories.Diagram.Equalizer.html#422" class="Generalizable">A</a> <a id="450" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="452" href="Categories.Diagram.Equalizer.html#424" class="Generalizable">B</a>
<a id="455" class="Keyword">record</a> <a id="IsEqualizer"></a><a id="462" href="Categories.Diagram.Equalizer.html#462" class="Record">IsEqualizer</a> <a id="474" class="Symbol">{</a><a id="475" href="Categories.Diagram.Equalizer.html#475" class="Bound">E</a><a id="476" class="Symbol">}</a> <a id="478" class="Symbol">(</a><a id="479" href="Categories.Diagram.Equalizer.html#479" class="Bound">arr</a> <a id="483" class="Symbol">:</a> <a id="485" href="Categories.Diagram.Equalizer.html#475" class="Bound">E</a> <a id="487" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="489" href="Categories.Diagram.Equalizer.html#422" class="Generalizable">A</a><a id="490" class="Symbol">)</a> <a id="492" class="Symbol">(</a><a id="493" href="Categories.Diagram.Equalizer.html#493" class="Bound">f</a> <a id="495" href="Categories.Diagram.Equalizer.html#495" class="Bound">g</a> <a id="497" class="Symbol">:</a> <a id="499" href="Categories.Diagram.Equalizer.html#422" class="Generalizable">A</a> <a id="501" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="503" href="Categories.Diagram.Equalizer.html#424" class="Generalizable">B</a><a id="504" class="Symbol">)</a> <a id="506" class="Symbol">:</a> <a id="508" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="512" class="Symbol">(</a><a id="513" href="Categories.Diagram.Equalizer.html#158" class="Bound">o</a> <a id="515" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="517" href="Categories.Diagram.Equalizer.html#160" class="Bound">ℓ</a> <a id="519" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="521" href="Categories.Diagram.Equalizer.html#162" class="Bound">e</a><a id="522" class="Symbol">)</a> <a id="524" class="Keyword">where</a>
<a id="532" class="Keyword">field</a>
<a id="IsEqualizer.equality"></a><a id="542" href="Categories.Diagram.Equalizer.html#542" class="Field">equality</a> <a id="551" class="Symbol">:</a> <a id="553" href="Categories.Diagram.Equalizer.html#493" class="Bound">f</a> <a id="555" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="557" href="Categories.Diagram.Equalizer.html#479" class="Bound">arr</a> <a id="561" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="563" href="Categories.Diagram.Equalizer.html#495" class="Bound">g</a> <a id="565" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="567" href="Categories.Diagram.Equalizer.html#479" class="Bound">arr</a>
<a id="IsEqualizer.equalize"></a><a id="575" href="Categories.Diagram.Equalizer.html#575" class="Field">equalize</a> <a id="584" class="Symbol">:</a> <a id="586" class="Symbol">∀</a> <a id="588" class="Symbol">{</a><a id="589" href="Categories.Diagram.Equalizer.html#589" class="Bound">h</a> <a id="591" class="Symbol">:</a> <a id="593" href="Categories.Diagram.Equalizer.html#426" class="Generalizable">X</a> <a id="595" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="597" href="Categories.Diagram.Equalizer.html#489" class="Bound">A</a><a id="598" class="Symbol">}</a> <a id="600" class="Symbol">→</a> <a id="602" href="Categories.Diagram.Equalizer.html#493" class="Bound">f</a> <a id="604" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="606" href="Categories.Diagram.Equalizer.html#589" class="Bound">h</a> <a id="608" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="610" href="Categories.Diagram.Equalizer.html#495" class="Bound">g</a> <a id="612" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="614" href="Categories.Diagram.Equalizer.html#589" class="Bound">h</a> <a id="616" class="Symbol">→</a> <a id="618" href="Categories.Diagram.Equalizer.html#426" class="Generalizable">X</a> <a id="620" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="622" href="Categories.Diagram.Equalizer.html#475" class="Bound">E</a>
<a id="IsEqualizer.universal"></a><a id="628" href="Categories.Diagram.Equalizer.html#628" class="Field">universal</a> <a id="638" class="Symbol">:</a> <a id="640" class="Symbol">∀</a> <a id="642" class="Symbol">{</a><a id="643" href="Categories.Diagram.Equalizer.html#643" class="Bound">eq</a> <a id="646" class="Symbol">:</a> <a id="648" href="Categories.Diagram.Equalizer.html#493" class="Bound">f</a> <a id="650" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="652" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a> <a id="654" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="656" href="Categories.Diagram.Equalizer.html#495" class="Bound">g</a> <a id="658" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="660" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a><a id="661" class="Symbol">}</a> <a id="663" class="Symbol">→</a> <a id="665" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a> <a id="667" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="669" href="Categories.Diagram.Equalizer.html#479" class="Bound">arr</a> <a id="673" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="675" href="Categories.Diagram.Equalizer.html#575" class="Field">equalize</a> <a id="684" href="Categories.Diagram.Equalizer.html#643" class="Bound">eq</a>
<a id="IsEqualizer.unique"></a><a id="691" href="Categories.Diagram.Equalizer.html#691" class="Field">unique</a> <a id="701" class="Symbol">:</a> <a id="703" class="Symbol">∀</a> <a id="705" class="Symbol">{</a><a id="706" href="Categories.Diagram.Equalizer.html#706" class="Bound">eq</a> <a id="709" class="Symbol">:</a> <a id="711" href="Categories.Diagram.Equalizer.html#493" class="Bound">f</a> <a id="713" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="715" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a> <a id="717" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="719" href="Categories.Diagram.Equalizer.html#495" class="Bound">g</a> <a id="721" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="723" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a><a id="724" class="Symbol">}</a> <a id="726" class="Symbol">→</a> <a id="728" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a> <a id="730" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="732" href="Categories.Diagram.Equalizer.html#479" class="Bound">arr</a> <a id="736" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="738" href="Categories.Diagram.Equalizer.html#440" class="Generalizable">i</a> <a id="740" class="Symbol">→</a> <a id="742" href="Categories.Diagram.Equalizer.html#440" class="Generalizable">i</a> <a id="744" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="746" href="Categories.Diagram.Equalizer.html#575" class="Field">equalize</a> <a id="755" href="Categories.Diagram.Equalizer.html#706" class="Bound">eq</a>
<a id="IsEqualizer.unique′"></a><a id="761" href="Categories.Diagram.Equalizer.html#761" class="Function">unique′</a> <a id="769" class="Symbol">:</a> <a id="771" class="Symbol">(</a><a id="772" href="Categories.Diagram.Equalizer.html#772" class="Bound">eq</a> <a id="775" href="Categories.Diagram.Equalizer.html#775" class="Bound">eq′</a> <a id="779" class="Symbol">:</a> <a id="781" href="Categories.Diagram.Equalizer.html#493" class="Bound">f</a> <a id="783" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="785" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a> <a id="787" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="789" href="Categories.Diagram.Equalizer.html#495" class="Bound">g</a> <a id="791" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="793" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a><a id="794" class="Symbol">)</a> <a id="796" class="Symbol">→</a> <a id="798" href="Categories.Diagram.Equalizer.html#575" class="Field">equalize</a> <a id="807" href="Categories.Diagram.Equalizer.html#772" class="Bound">eq</a> <a id="810" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="812" href="Categories.Diagram.Equalizer.html#575" class="Field">equalize</a> <a id="821" href="Categories.Diagram.Equalizer.html#775" class="Bound">eq′</a>
<a id="827" href="Categories.Diagram.Equalizer.html#761" class="Function">unique′</a> <a id="835" href="Categories.Diagram.Equalizer.html#835" class="Bound">eq</a> <a id="838" href="Categories.Diagram.Equalizer.html#838" class="Bound">eq′</a> <a id="842" class="Symbol">=</a> <a id="844" href="Categories.Diagram.Equalizer.html#691" class="Field">unique</a> <a id="851" href="Categories.Diagram.Equalizer.html#628" class="Field">universal</a>
<a id="IsEqualizer.id-equalize"></a><a id="864" href="Categories.Diagram.Equalizer.html#864" class="Function">id-equalize</a> <a id="876" class="Symbol">:</a> <a id="878" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="881" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="883" href="Categories.Diagram.Equalizer.html#575" class="Field">equalize</a> <a id="892" href="Categories.Diagram.Equalizer.html#542" class="Field">equality</a>
<a id="903" href="Categories.Diagram.Equalizer.html#864" class="Function">id-equalize</a> <a id="915" class="Symbol">=</a> <a id="917" href="Categories.Diagram.Equalizer.html#691" class="Field">unique</a> <a id="924" class="Symbol">(</a><a id="925" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="929" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a><a id="938" class="Symbol">)</a>
<a id="IsEqualizer.equalize-resp-≈"></a><a id="943" href="Categories.Diagram.Equalizer.html#943" class="Function">equalize-resp-≈</a> <a id="959" class="Symbol">:</a> <a id="961" class="Symbol">∀</a> <a id="963" class="Symbol">{</a><a id="964" href="Categories.Diagram.Equalizer.html#964" class="Bound">eq</a> <a id="967" class="Symbol">:</a> <a id="969" href="Categories.Diagram.Equalizer.html#493" class="Bound">f</a> <a id="971" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="973" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a> <a id="975" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="977" href="Categories.Diagram.Equalizer.html#495" class="Bound">g</a> <a id="979" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="981" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a><a id="982" class="Symbol">}</a> <a id="984" class="Symbol">{</a><a id="985" href="Categories.Diagram.Equalizer.html#985" class="Bound">eq′</a> <a id="989" class="Symbol">:</a> <a id="991" href="Categories.Diagram.Equalizer.html#493" class="Bound">f</a> <a id="993" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="995" href="Categories.Diagram.Equalizer.html#440" class="Generalizable">i</a> <a id="997" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="999" href="Categories.Diagram.Equalizer.html#495" class="Bound">g</a> <a id="1001" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1003" href="Categories.Diagram.Equalizer.html#440" class="Generalizable">i</a><a id="1004" class="Symbol">}</a> <a id="1006" class="Symbol">→</a>
<a id="1012" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a> <a id="1014" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1016" href="Categories.Diagram.Equalizer.html#440" class="Generalizable">i</a> <a id="1018" class="Symbol">→</a> <a id="1020" href="Categories.Diagram.Equalizer.html#575" class="Field">equalize</a> <a id="1029" href="Categories.Diagram.Equalizer.html#964" class="Bound">eq</a> <a id="1032" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1034" href="Categories.Diagram.Equalizer.html#575" class="Field">equalize</a> <a id="1043" href="Categories.Diagram.Equalizer.html#985" class="Bound">eq′</a>
<a id="1049" href="Categories.Diagram.Equalizer.html#943" class="Function">equalize-resp-≈</a> <a id="1065" class="Symbol">{</a><a id="1066" class="Argument">h</a> <a id="1068" class="Symbol">=</a> <a id="1070" href="Categories.Diagram.Equalizer.html#1070" class="Bound">h</a><a id="1071" class="Symbol">}</a> <a id="1073" class="Symbol">{</a><a id="1074" class="Argument">i</a> <a id="1076" class="Symbol">=</a> <a id="1078" href="Categories.Diagram.Equalizer.html#1078" class="Bound">i</a><a id="1079" class="Symbol">}</a> <a id="1081" class="Symbol">{</a><a id="1082" class="Argument">eq</a> <a id="1085" class="Symbol">=</a> <a id="1087" href="Categories.Diagram.Equalizer.html#1087" class="Bound">eq</a><a id="1089" class="Symbol">}</a> <a id="1091" class="Symbol">{</a><a id="1092" class="Argument">eq′</a> <a id="1096" class="Symbol">=</a> <a id="1098" href="Categories.Diagram.Equalizer.html#1098" class="Bound">eq′</a><a id="1101" class="Symbol">}</a> <a id="1103" href="Categories.Diagram.Equalizer.html#1103" class="Bound">h≈i</a> <a id="1107" class="Symbol">=</a> <a id="1109" href="Categories.Diagram.Equalizer.html#691" class="Field">unique</a> <a id="1116" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="1118" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="1128" href="Categories.Diagram.Equalizer.html#1078" class="Bound">i</a> <a id="1146" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="1150" href="Categories.Diagram.Equalizer.html#1103" class="Bound">h≈i</a> <a id="1154" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="1160" href="Categories.Diagram.Equalizer.html#1070" class="Bound">h</a> <a id="1178" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1181" href="Categories.Diagram.Equalizer.html#628" class="Field">universal</a> <a id="1191" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1197" href="Categories.Diagram.Equalizer.html#479" class="Bound">arr</a> <a id="1201" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1203" href="Categories.Diagram.Equalizer.html#575" class="Field">equalize</a> <a id="1212" href="Categories.Diagram.Equalizer.html#1087" class="Bound">eq</a> <a id="1215" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="IsEqualizer.equalize-resp-≈′"></a><a id="1220" href="Categories.Diagram.Equalizer.html#1220" class="Function">equalize-resp-≈′</a> <a id="1237" class="Symbol">:</a> <a id="1239" class="Symbol">(</a><a id="1240" href="Categories.Diagram.Equalizer.html#1240" class="Bound">eq</a> <a id="1243" class="Symbol">:</a> <a id="1245" href="Categories.Diagram.Equalizer.html#493" class="Bound">f</a> <a id="1247" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1249" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a> <a id="1251" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1253" href="Categories.Diagram.Equalizer.html#495" class="Bound">g</a> <a id="1255" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1257" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a><a id="1258" class="Symbol">)</a> <a id="1260" class="Symbol">→</a> <a id="1262" class="Symbol">(</a><a id="1263" href="Categories.Diagram.Equalizer.html#1263" class="Bound">eq′</a> <a id="1267" class="Symbol">:</a> <a id="1269" href="Categories.Diagram.Equalizer.html#493" class="Bound">f</a> <a id="1271" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1273" href="Categories.Diagram.Equalizer.html#440" class="Generalizable">i</a> <a id="1275" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1277" href="Categories.Diagram.Equalizer.html#495" class="Bound">g</a> <a id="1279" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1281" href="Categories.Diagram.Equalizer.html#440" class="Generalizable">i</a><a id="1282" class="Symbol">)</a> <a id="1284" class="Symbol">→</a>
<a id="1290" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a> <a id="1292" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1294" href="Categories.Diagram.Equalizer.html#440" class="Generalizable">i</a> <a id="1296" class="Symbol">→</a> <a id="1298" href="Categories.Diagram.Equalizer.html#442" class="Generalizable">j</a> <a id="1300" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1302" href="Categories.Diagram.Equalizer.html#575" class="Field">equalize</a> <a id="1311" href="Categories.Diagram.Equalizer.html#1240" class="Bound">eq</a> <a id="1314" class="Symbol">→</a> <a id="1316" href="Categories.Diagram.Equalizer.html#444" class="Generalizable">k</a> <a id="1318" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1320" href="Categories.Diagram.Equalizer.html#575" class="Field">equalize</a> <a id="1329" href="Categories.Diagram.Equalizer.html#1263" class="Bound">eq′</a> <a id="1333" class="Symbol">→</a> <a id="1335" href="Categories.Diagram.Equalizer.html#442" class="Generalizable">j</a> <a id="1337" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1339" href="Categories.Diagram.Equalizer.html#444" class="Generalizable">k</a>
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<a id="1407" href="Categories.Diagram.Equalizer.html#1365" class="Bound">j</a> <a id="1420" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1423" href="Categories.Diagram.Equalizer.html#1387" class="Bound">eqj</a> <a id="1427" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1433" href="Categories.Diagram.Equalizer.html#575" class="Field">equalize</a> <a id="1442" href="Categories.Diagram.Equalizer.html#1376" class="Bound">eq</a> <a id="1446" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1449" href="Categories.Diagram.Equalizer.html#943" class="Function">equalize-resp-≈</a> <a id="1465" href="Categories.Diagram.Equalizer.html#1383" class="Bound">h≈i</a> <a id="1469" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="1502" href="Categories.Diagram.Equalizer.html#1373" class="Bound">k</a> <a id="1515" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
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<a id="1664" href="Categories.Diagram.Equalizer.html#575" class="Field">equalize</a> <a id="1673" class="Symbol">(</a><a id="1674" href="Categories.Morphism.Reasoning.Core.html#3587" class="Function">extendʳ</a> <a id="1682" href="Categories.Diagram.Equalizer.html#542" class="Field">equality</a><a id="1690" class="Symbol">)</a> <a id="1692" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="1696" href="Categories.Diagram.Equalizer.html#691" class="Field">unique</a> <a id="1703" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="1708" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
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<a id="1915" class="Keyword">where</a> <a id="1921" class="Keyword">open</a> <a id="1926" href="Categories.Diagram.Equalizer.html#462" class="Module">IsEqualizer</a> <a id="1938" href="Categories.Diagram.Equalizer.html#1814" class="Bound">e</a>
<a id="1941" class="Keyword">record</a> <a id="Equalizer"></a><a id="1948" href="Categories.Diagram.Equalizer.html#1948" class="Record">Equalizer</a> <a id="1958" class="Symbol">(</a><a id="1959" href="Categories.Diagram.Equalizer.html#1959" class="Bound">f</a> <a id="1961" href="Categories.Diagram.Equalizer.html#1961" class="Bound">g</a> <a id="1963" class="Symbol">:</a> <a id="1965" href="Categories.Diagram.Equalizer.html#422" class="Generalizable">A</a> <a id="1967" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="1969" href="Categories.Diagram.Equalizer.html#424" class="Generalizable">B</a><a id="1970" class="Symbol">)</a> <a id="1972" class="Symbol">:</a> <a id="1974" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1978" class="Symbol">(</a><a id="1979" href="Categories.Diagram.Equalizer.html#158" class="Bound">o</a> <a id="1981" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1983" href="Categories.Diagram.Equalizer.html#160" class="Bound">ℓ</a> <a id="1985" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1987" href="Categories.Diagram.Equalizer.html#162" class="Bound">e</a><a id="1988" class="Symbol">)</a> <a id="1990" class="Keyword">where</a>
<a id="1998" class="Keyword">field</a>
<a id="2008" class="Symbol">{</a><a id="Equalizer.obj"></a><a id="2009" href="Categories.Diagram.Equalizer.html#2009" class="Field">obj</a><a id="2012" class="Symbol">}</a> <a id="2014" class="Symbol">:</a> <a id="2016" href="Categories.Category.Core.html#559" class="Field">Obj</a>
<a id="Equalizer.arr"></a><a id="2024" href="Categories.Diagram.Equalizer.html#2024" class="Field">arr</a> <a id="2030" class="Symbol">:</a> <a id="2032" href="Categories.Diagram.Equalizer.html#2009" class="Field">obj</a> <a id="2036" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="2038" href="Categories.Diagram.Equalizer.html#1965" class="Bound">A</a>
<a id="Equalizer.isEqualizer"></a><a id="2044" href="Categories.Diagram.Equalizer.html#2044" class="Field">isEqualizer</a> <a id="2056" class="Symbol">:</a> <a id="2058" href="Categories.Diagram.Equalizer.html#462" class="Record">IsEqualizer</a> <a id="2070" href="Categories.Diagram.Equalizer.html#2024" class="Field">arr</a> <a id="2074" href="Categories.Diagram.Equalizer.html#1959" class="Bound">f</a> <a id="2076" href="Categories.Diagram.Equalizer.html#1961" class="Bound">g</a>
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<a id="Equalizer⇒Mono"></a><a id="2118" href="Categories.Diagram.Equalizer.html#2118" class="Function">Equalizer⇒Mono</a> <a id="2133" class="Symbol">:</a> <a id="2135" class="Symbol">(</a><a id="2136" href="Categories.Diagram.Equalizer.html#2136" class="Bound">e</a> <a id="2138" class="Symbol">:</a> <a id="2140" href="Categories.Diagram.Equalizer.html#1948" class="Record">Equalizer</a> <a id="2150" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a> <a id="2152" href="Categories.Diagram.Equalizer.html#440" class="Generalizable">i</a><a id="2153" class="Symbol">)</a> <a id="2155" class="Symbol">→</a> <a id="2157" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="2162" class="Symbol">(</a><a id="2163" href="Categories.Diagram.Equalizer.html#2024" class="Field">Equalizer.arr</a> <a id="2177" href="Categories.Diagram.Equalizer.html#2136" class="Bound">e</a><a id="2178" class="Symbol">)</a>
<a id="2180" href="Categories.Diagram.Equalizer.html#2118" class="Function">Equalizer⇒Mono</a> <a id="2195" href="Categories.Diagram.Equalizer.html#2195" class="Bound">e</a> <a id="2197" class="Symbol">=</a> <a id="2199" href="Categories.Diagram.Equalizer.html#1745" class="Function">IsEqualizer⇒Mono</a> <a id="2216" href="Categories.Diagram.Equalizer.html#2044" class="Function">isEqualizer</a>
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<a id="2486" class="Symbol">}</a>
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<a id="2524" href="Categories.Diagram.Equalizer.html#2524" class="Function">repack</a> <a id="2531" class="Symbol">:</a> <a id="2533" class="Symbol">(</a><a id="2534" href="Categories.Diagram.Equalizer.html#2534" class="Bound">e₁</a> <a id="2537" href="Categories.Diagram.Equalizer.html#2537" class="Bound">e₂</a> <a id="2540" class="Symbol">:</a> <a id="2542" href="Categories.Diagram.Equalizer.html#1948" class="Record">Equalizer</a> <a id="2552" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a> <a id="2554" href="Categories.Diagram.Equalizer.html#440" class="Generalizable">i</a><a id="2555" class="Symbol">)</a> <a id="2557" class="Symbol">→</a> <a id="2559" href="Categories.Diagram.Equalizer.html#2009" class="Field">obj</a> <a id="2563" href="Categories.Diagram.Equalizer.html#2534" class="Bound">e₁</a> <a id="2566" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="2568" href="Categories.Diagram.Equalizer.html#2009" class="Field">obj</a> <a id="2572" href="Categories.Diagram.Equalizer.html#2537" class="Bound">e₂</a>
<a id="2579" href="Categories.Diagram.Equalizer.html#2524" class="Function">repack</a> <a id="2586" href="Categories.Diagram.Equalizer.html#2586" class="Bound">e₁</a> <a id="2589" href="Categories.Diagram.Equalizer.html#2589" class="Bound">e₂</a> <a id="2592" class="Symbol">=</a> <a id="2594" href="Categories.Diagram.Equalizer.html#575" class="Function">equalize</a> <a id="2603" href="Categories.Diagram.Equalizer.html#2589" class="Bound">e₂</a> <a id="2606" class="Symbol">(</a><a id="2607" href="Categories.Diagram.Equalizer.html#542" class="Function">equality</a> <a id="2616" href="Categories.Diagram.Equalizer.html#2586" class="Bound">e₁</a><a id="2618" class="Symbol">)</a>
<a id="2625" href="Categories.Diagram.Equalizer.html#2625" class="Function">repack∘</a> <a id="2633" class="Symbol">:</a> <a id="2635" class="Symbol">(</a><a id="2636" href="Categories.Diagram.Equalizer.html#2636" class="Bound">e₁</a> <a id="2639" href="Categories.Diagram.Equalizer.html#2639" class="Bound">e₂</a> <a id="2642" href="Categories.Diagram.Equalizer.html#2642" class="Bound">e₃</a> <a id="2645" class="Symbol">:</a> <a id="2647" href="Categories.Diagram.Equalizer.html#1948" class="Record">Equalizer</a> <a id="2657" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a> <a id="2659" href="Categories.Diagram.Equalizer.html#440" class="Generalizable">i</a><a id="2660" class="Symbol">)</a> <a id="2662" class="Symbol">→</a> <a id="2664" href="Categories.Diagram.Equalizer.html#2524" class="Function">repack</a> <a id="2671" href="Categories.Diagram.Equalizer.html#2639" class="Bound">e₂</a> <a id="2674" href="Categories.Diagram.Equalizer.html#2642" class="Bound">e₃</a> <a id="2677" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2679" href="Categories.Diagram.Equalizer.html#2524" class="Function">repack</a> <a id="2686" href="Categories.Diagram.Equalizer.html#2636" class="Bound">e₁</a> <a id="2689" href="Categories.Diagram.Equalizer.html#2639" class="Bound">e₂</a> <a id="2692" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="2694" href="Categories.Diagram.Equalizer.html#2524" class="Function">repack</a> <a id="2701" href="Categories.Diagram.Equalizer.html#2636" class="Bound">e₁</a> <a id="2704" href="Categories.Diagram.Equalizer.html#2642" class="Bound">e₃</a>
<a id="2711" href="Categories.Diagram.Equalizer.html#2625" class="Function">repack∘</a> <a id="2719" href="Categories.Diagram.Equalizer.html#2719" class="Bound">e₁</a> <a id="2722" href="Categories.Diagram.Equalizer.html#2722" class="Bound">e₂</a> <a id="2725" href="Categories.Diagram.Equalizer.html#2725" class="Bound">e₃</a> <a id="2728" class="Symbol">=</a> <a id="2730" href="Categories.Diagram.Equalizer.html#691" class="Function">unique</a> <a id="2737" href="Categories.Diagram.Equalizer.html#2725" class="Bound">e₃</a> <a id="2740" class="Symbol">(</a><a id="2741" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="2743" class="Symbol">(</a><a id="2744" href="Categories.Morphism.Reasoning.Core.html#6062" class="Function">glueTrianglesʳ</a> <a id="2759" class="Symbol">(</a><a id="2760" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="2762" class="Symbol">(</a><a id="2763" href="Categories.Diagram.Equalizer.html#628" class="Function">universal</a> <a id="2773" href="Categories.Diagram.Equalizer.html#2725" class="Bound">e₃</a><a id="2775" class="Symbol">))</a> <a id="2778" class="Symbol">(</a><a id="2779" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="2781" class="Symbol">(</a><a id="2782" href="Categories.Diagram.Equalizer.html#628" class="Function">universal</a> <a id="2792" href="Categories.Diagram.Equalizer.html#2722" class="Bound">e₂</a><a id="2794" class="Symbol">))))</a>
<a id="2804" href="Categories.Diagram.Equalizer.html#2804" class="Function">repack-cancel</a> <a id="2818" class="Symbol">:</a> <a id="2820" class="Symbol">(</a><a id="2821" href="Categories.Diagram.Equalizer.html#2821" class="Bound">e₁</a> <a id="2824" href="Categories.Diagram.Equalizer.html#2824" class="Bound">e₂</a> <a id="2827" class="Symbol">:</a> <a id="2829" href="Categories.Diagram.Equalizer.html#1948" class="Record">Equalizer</a> <a id="2839" href="Categories.Diagram.Equalizer.html#438" class="Generalizable">h</a> <a id="2841" href="Categories.Diagram.Equalizer.html#440" class="Generalizable">i</a><a id="2842" class="Symbol">)</a> <a id="2844" class="Symbol">→</a> <a id="2846" href="Categories.Diagram.Equalizer.html#2524" class="Function">repack</a> <a id="2853" href="Categories.Diagram.Equalizer.html#2821" class="Bound">e₁</a> <a id="2856" href="Categories.Diagram.Equalizer.html#2824" class="Bound">e₂</a> <a id="2859" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2861" href="Categories.Diagram.Equalizer.html#2524" class="Function">repack</a> <a id="2868" href="Categories.Diagram.Equalizer.html#2824" class="Bound">e₂</a> <a id="2871" href="Categories.Diagram.Equalizer.html#2821" class="Bound">e₁</a> <a id="2874" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="2876" href="Categories.Category.Core.html#630" class="Field">id</a>
<a id="2883" href="Categories.Diagram.Equalizer.html#2804" class="Function">repack-cancel</a> <a id="2897" href="Categories.Diagram.Equalizer.html#2897" class="Bound">e₁</a> <a id="2900" href="Categories.Diagram.Equalizer.html#2900" class="Bound">e₂</a> <a id="2903" class="Symbol">=</a> <a id="2905" href="Categories.Diagram.Equalizer.html#2625" class="Function">repack∘</a> <a id="2913" href="Categories.Diagram.Equalizer.html#2900" class="Bound">e₂</a> <a id="2916" href="Categories.Diagram.Equalizer.html#2897" class="Bound">e₁</a> <a id="2919" href="Categories.Diagram.Equalizer.html#2900" class="Bound">e₂</a> <a id="2922" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2924" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="2926" class="Symbol">(</a><a id="2927" href="Categories.Diagram.Equalizer.html#864" class="Function">id-equalize</a> <a id="2939" href="Categories.Diagram.Equalizer.html#2900" class="Bound">e₂</a><a id="2941" class="Symbol">)</a>
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