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<html><head><meta charset="utf-8"><title>Categories.Morphism.Properties</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>
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<a id="69" class="Keyword">module</a> <a id="76" href="Categories.Morphism.Properties.html" class="Module">Categories.Morphism.Properties</a> <a id="107" class="Symbol">{</a><a id="108" href="Categories.Morphism.Properties.html#108" class="Bound">o</a> <a id="110" href="Categories.Morphism.Properties.html#110" class="Bound">ℓ</a> <a id="112" href="Categories.Morphism.Properties.html#112" class="Bound">e</a><a id="113" class="Symbol">}</a> <a id="115" class="Symbol">(</a><a id="116" href="Categories.Morphism.Properties.html#116" class="Bound">𝒞</a> <a id="118" class="Symbol">:</a> <a id="120" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="129" href="Categories.Morphism.Properties.html#108" class="Bound">o</a> <a id="131" href="Categories.Morphism.Properties.html#110" class="Bound">ℓ</a> <a id="133" href="Categories.Morphism.Properties.html#112" class="Bound">e</a><a id="134" class="Symbol">)</a> <a id="136" class="Keyword">where</a>
<a id="143" class="Keyword">open</a> <a id="148" class="Keyword">import</a> <a id="155" href="Function.Base.html" class="Module">Function.Base</a> <a id="169" class="Keyword">using</a> <a id="175" class="Symbol">(</a><a id="176" href="Function.Base.html#1928" class="Function Operator">_$_</a><a id="179" class="Symbol">)</a>
<a id="181" class="Keyword">open</a> <a id="186" class="Keyword">import</a> <a id="193" href="Data.Product.html" class="Module">Data.Product</a> <a id="206" class="Keyword">using</a> <a id="212" class="Symbol">(</a><a id="213" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">_,_</a><a id="216" class="Symbol">;</a> <a id="218" href="Data.Product.html#1176" class="Function Operator">_×_</a><a id="221" class="Symbol">)</a>
<a id="224" class="Keyword">open</a> <a id="229" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="238" href="Categories.Morphism.Properties.html#116" class="Bound">𝒞</a>
<a id="240" class="Keyword">open</a> <a id="245" href="Categories.Category.html#824" class="Module">Definitions</a> <a id="257" href="Categories.Morphism.Properties.html#116" class="Bound">𝒞</a>
<a id="259" class="Keyword">open</a> <a id="264" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="278" class="Keyword">import</a> <a id="285" href="Categories.Morphism.html" class="Module">Categories.Morphism</a> <a id="305" class="Symbol">as</a> <a id="308" class="Module">M</a>
<a id="310" class="Keyword">open</a> <a id="315" href="Categories.Morphism.html" class="Module">M</a> <a id="317" href="Categories.Morphism.Properties.html#116" class="Bound">𝒞</a>
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<a id="364" class="Keyword">private</a>
<a id="374" class="Keyword">variable</a>
<a id="387" href="Categories.Morphism.Properties.html#387" class="Generalizable">A</a> <a id="389" href="Categories.Morphism.Properties.html#389" class="Generalizable">B</a> <a id="391" href="Categories.Morphism.Properties.html#391" class="Generalizable">C</a> <a id="393" href="Categories.Morphism.Properties.html#393" class="Generalizable">D</a> <a id="395" class="Symbol">:</a> <a id="397" href="Categories.Category.Core.html#559" class="Field">Obj</a>
<a id="405" href="Categories.Morphism.Properties.html#405" class="Generalizable">f</a> <a id="407" href="Categories.Morphism.Properties.html#407" class="Generalizable">g</a> <a id="409" href="Categories.Morphism.Properties.html#409" class="Generalizable">h</a> <a id="411" href="Categories.Morphism.Properties.html#411" class="Generalizable">i</a> <a id="413" class="Symbol">:</a> <a id="415" href="Categories.Morphism.Properties.html#387" class="Generalizable">A</a> <a id="417" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="419" href="Categories.Morphism.Properties.html#389" class="Generalizable">B</a>
<a id="422" class="Keyword">module</a> <a id="429" href="Categories.Morphism.Properties.html#429" class="Module">_</a> <a id="431" class="Symbol">(</a><a id="432" href="Categories.Morphism.Properties.html#432" class="Bound">iso</a> <a id="436" class="Symbol">:</a> <a id="438" href="Categories.Morphism.html#1528" class="Record">Iso</a> <a id="442" href="Categories.Morphism.Properties.html#405" class="Generalizable">f</a> <a id="444" href="Categories.Morphism.Properties.html#407" class="Generalizable">g</a><a id="445" class="Symbol">)</a> <a id="447" class="Keyword">where</a>
<a id="456" class="Keyword">open</a> <a id="461" href="Categories.Morphism.html#1528" class="Module">Iso</a> <a id="465" href="Categories.Morphism.Properties.html#432" class="Bound">iso</a>
<a id="472" href="Categories.Morphism.Properties.html#472" class="Function">Iso-resp-≈</a> <a id="483" class="Symbol">:</a> <a id="485" href="Categories.Morphism.Properties.html#442" class="Bound">f</a> <a id="487" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="489" href="Categories.Morphism.Properties.html#409" class="Generalizable">h</a> <a id="491" class="Symbol">→</a> <a id="493" href="Categories.Morphism.Properties.html#444" class="Bound">g</a> <a id="495" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="497" href="Categories.Morphism.Properties.html#411" class="Generalizable">i</a> <a id="499" class="Symbol">→</a> <a id="501" href="Categories.Morphism.html#1528" class="Record">Iso</a> <a id="505" href="Categories.Morphism.Properties.html#409" class="Generalizable">h</a> <a id="507" href="Categories.Morphism.Properties.html#411" class="Generalizable">i</a>
<a id="511" href="Categories.Morphism.Properties.html#472" class="Function">Iso-resp-≈</a> <a id="522" class="Symbol">{</a><a id="523" class="Argument">h</a> <a id="525" class="Symbol">=</a> <a id="527" href="Categories.Morphism.Properties.html#527" class="Bound">h</a><a id="528" class="Symbol">}</a> <a id="530" class="Symbol">{</a><a id="531" class="Argument">i</a> <a id="533" class="Symbol">=</a> <a id="535" href="Categories.Morphism.Properties.html#535" class="Bound">i</a><a id="536" class="Symbol">}</a> <a id="538" href="Categories.Morphism.Properties.html#538" class="Bound">eq₁</a> <a id="542" href="Categories.Morphism.Properties.html#542" class="Bound">eq₂</a> <a id="546" class="Symbol">=</a> <a id="548" class="Keyword">record</a>
<a id="559" class="Symbol">{</a> <a id="561" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="566" class="Symbol">=</a> <a id="568" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="580" href="Categories.Morphism.Properties.html#535" class="Bound">i</a> <a id="582" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="584" href="Categories.Morphism.Properties.html#527" class="Bound">h</a> <a id="586" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="590" href="Categories.Morphism.Properties.html#542" class="Bound">eq₂</a> <a id="594" href="Categories.Category.Core.html#2634" class="Function Operator">⟩∘⟨</a> <a id="598" href="Categories.Morphism.Properties.html#538" class="Bound">eq₁</a> <a id="602" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="610" href="Categories.Morphism.Properties.html#444" class="Bound">g</a> <a id="612" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="614" href="Categories.Morphism.Properties.html#442" class="Bound">f</a> <a id="616" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="619" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="624" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="632" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="638" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="644" class="Symbol">;</a> <a id="646" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="651" class="Symbol">=</a> <a id="653" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="665" href="Categories.Morphism.Properties.html#527" class="Bound">h</a> <a id="667" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="669" href="Categories.Morphism.Properties.html#535" class="Bound">i</a> <a id="671" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="675" href="Categories.Morphism.Properties.html#538" class="Bound">eq₁</a> <a id="679" href="Categories.Category.Core.html#2634" class="Function Operator">⟩∘⟨</a> <a id="683" href="Categories.Morphism.Properties.html#542" class="Bound">eq₂</a> <a id="687" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="695" href="Categories.Morphism.Properties.html#442" class="Bound">f</a> <a id="697" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="699" href="Categories.Morphism.Properties.html#444" class="Bound">g</a> <a id="701" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="704" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="709" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="717" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="723" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="729" class="Symbol">}</a>
<a id="734" href="Categories.Morphism.Properties.html#734" class="Function">Iso-swap</a> <a id="743" class="Symbol">:</a> <a id="745" href="Categories.Morphism.html#1528" class="Record">Iso</a> <a id="749" href="Categories.Morphism.Properties.html#444" class="Bound">g</a> <a id="751" href="Categories.Morphism.Properties.html#442" class="Bound">f</a>
<a id="755" href="Categories.Morphism.Properties.html#734" class="Function">Iso-swap</a> <a id="764" class="Symbol">=</a> <a id="766" class="Keyword">record</a>
<a id="777" class="Symbol">{</a> <a id="779" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="784" class="Symbol">=</a> <a id="786" href="Categories.Morphism.html#1612" class="Field">isoʳ</a>
<a id="795" class="Symbol">;</a> <a id="797" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="802" class="Symbol">=</a> <a id="804" href="Categories.Morphism.html#1586" class="Field">isoˡ</a>
<a id="813" class="Symbol">}</a>
<a id="818" href="Categories.Morphism.Properties.html#818" class="Function">Iso⇒Mono</a> <a id="827" class="Symbol">:</a> <a id="829" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="834" href="Categories.Morphism.Properties.html#442" class="Bound">f</a>
<a id="838" href="Categories.Morphism.Properties.html#818" class="Function">Iso⇒Mono</a> <a id="847" href="Categories.Morphism.Properties.html#847" class="Bound">h</a> <a id="849" href="Categories.Morphism.Properties.html#849" class="Bound">i</a> <a id="851" href="Categories.Morphism.Properties.html#851" class="Bound">eq</a> <a id="854" class="Symbol">=</a> <a id="856" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="866" href="Categories.Morphism.Properties.html#847" class="Bound">h</a> <a id="878" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="881" href="Categories.Morphism.Reasoning.Core.html#3063" class="Function">introˡ</a> <a id="888" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="893" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="929" href="Categories.Morphism.Properties.html#444" class="Bound">g</a> <a id="931" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="933" href="Categories.Morphism.Properties.html#442" class="Bound">f</a> <a id="935" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="937" href="Categories.Morphism.Properties.html#849" class="Bound">i</a> <a id="941" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="944" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="952" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="957" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="963" href="Categories.Morphism.Properties.html#849" class="Bound">i</a> <a id="975" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="980" href="Categories.Morphism.Properties.html#980" class="Function">Iso⇒Epi</a> <a id="988" class="Symbol">:</a> <a id="990" href="Categories.Morphism.html#875" class="Function">Epi</a> <a id="994" href="Categories.Morphism.Properties.html#442" class="Bound">f</a>
<a id="998" href="Categories.Morphism.Properties.html#980" class="Function">Iso⇒Epi</a> <a id="1006" href="Categories.Morphism.Properties.html#1006" class="Bound">h</a> <a id="1008" href="Categories.Morphism.Properties.html#1008" class="Bound">i</a> <a id="1010" href="Categories.Morphism.Properties.html#1010" class="Bound">eq</a> <a id="1013" class="Symbol">=</a> <a id="1015" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="1025" href="Categories.Morphism.Properties.html#1006" class="Bound">h</a> <a id="1037" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1040" href="Categories.Morphism.Reasoning.Core.html#2899" class="Function">introʳ</a> <a id="1047" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="1052" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1058" href="Categories.Morphism.Properties.html#1006" class="Bound">h</a> <a id="1060" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1062" href="Categories.Morphism.Properties.html#442" class="Bound">f</a> <a id="1064" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1066" href="Categories.Morphism.Properties.html#444" class="Bound">g</a> <a id="1070" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1073" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="1079" href="Categories.Morphism.Properties.html#1010" class="Bound">eq</a> <a id="1082" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1088" class="Symbol">(</a><a id="1089" href="Categories.Morphism.Properties.html#1008" class="Bound">i</a> <a id="1091" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1093" href="Categories.Morphism.Properties.html#442" class="Bound">f</a><a id="1094" class="Symbol">)</a> <a id="1096" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1098" href="Categories.Morphism.Properties.html#444" class="Bound">g</a> <a id="1100" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1103" href="Categories.Morphism.Reasoning.Core.html#6665" class="Function">cancelʳ</a> <a id="1111" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="1116" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1122" href="Categories.Morphism.Properties.html#1008" class="Bound">i</a> <a id="1134" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="Iso-∘"></a><a id="1137" href="Categories.Morphism.Properties.html#1137" class="Function">Iso-∘</a> <a id="1143" class="Symbol">:</a> <a id="1145" href="Categories.Morphism.html#1528" class="Record">Iso</a> <a id="1149" href="Categories.Morphism.Properties.html#405" class="Generalizable">f</a> <a id="1151" href="Categories.Morphism.Properties.html#407" class="Generalizable">g</a> <a id="1153" class="Symbol">→</a> <a id="1155" href="Categories.Morphism.html#1528" class="Record">Iso</a> <a id="1159" href="Categories.Morphism.Properties.html#409" class="Generalizable">h</a> <a id="1161" href="Categories.Morphism.Properties.html#411" class="Generalizable">i</a> <a id="1163" class="Symbol">→</a> <a id="1165" href="Categories.Morphism.html#1528" class="Record">Iso</a> <a id="1169" class="Symbol">(</a><a id="1170" href="Categories.Morphism.Properties.html#409" class="Generalizable">h</a> <a id="1172" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1174" href="Categories.Morphism.Properties.html#405" class="Generalizable">f</a><a id="1175" class="Symbol">)</a> <a id="1177" class="Symbol">(</a><a id="1178" href="Categories.Morphism.Properties.html#407" class="Generalizable">g</a> <a id="1180" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1182" href="Categories.Morphism.Properties.html#411" class="Generalizable">i</a><a id="1183" class="Symbol">)</a>
<a id="1185" href="Categories.Morphism.Properties.html#1137" class="Function">Iso-∘</a> <a id="1191" class="Symbol">{</a><a id="1192" class="Argument">f</a> <a id="1194" class="Symbol">=</a> <a id="1196" href="Categories.Morphism.Properties.html#1196" class="Bound">f</a><a id="1197" class="Symbol">}</a> <a id="1199" class="Symbol">{</a><a id="1200" class="Argument">g</a> <a id="1202" class="Symbol">=</a> <a id="1204" href="Categories.Morphism.Properties.html#1204" class="Bound">g</a><a id="1205" class="Symbol">}</a> <a id="1207" class="Symbol">{</a><a id="1208" class="Argument">h</a> <a id="1210" class="Symbol">=</a> <a id="1212" href="Categories.Morphism.Properties.html#1212" class="Bound">h</a><a id="1213" class="Symbol">}</a> <a id="1215" class="Symbol">{</a><a id="1216" class="Argument">i</a> <a id="1218" class="Symbol">=</a> <a id="1220" href="Categories.Morphism.Properties.html#1220" class="Bound">i</a><a id="1221" class="Symbol">}</a> <a id="1223" href="Categories.Morphism.Properties.html#1223" class="Bound">iso</a> <a id="1227" href="Categories.Morphism.Properties.html#1227" class="Bound">iso′</a> <a id="1232" class="Symbol">=</a> <a id="1234" class="Keyword">record</a>
<a id="1243" class="Symbol">{</a> <a id="1245" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="1250" class="Symbol">=</a> <a id="1252" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="1262" class="Symbol">(</a><a id="1263" href="Categories.Morphism.Properties.html#1204" class="Bound">g</a> <a id="1265" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1267" href="Categories.Morphism.Properties.html#1220" class="Bound">i</a><a id="1268" class="Symbol">)</a> <a id="1270" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1272" href="Categories.Morphism.Properties.html#1212" class="Bound">h</a> <a id="1274" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1276" href="Categories.Morphism.Properties.html#1196" class="Bound">f</a> <a id="1278" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1281" href="Categories.Morphism.Reasoning.Core.html#7037" class="Function">cancelInner</a> <a id="1293" class="Symbol">(</a><a id="1294" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="1299" href="Categories.Morphism.Properties.html#1227" class="Bound">iso′</a><a id="1303" class="Symbol">)</a> <a id="1305" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1311" href="Categories.Morphism.Properties.html#1204" class="Bound">g</a> <a id="1313" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1315" href="Categories.Morphism.Properties.html#1196" class="Bound">f</a> <a id="1327" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1330" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="1335" href="Categories.Morphism.Properties.html#1223" class="Bound">iso</a> <a id="1339" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1345" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="1361" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="1365" class="Symbol">;</a> <a id="1367" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="1372" class="Symbol">=</a> <a id="1374" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="1384" class="Symbol">(</a><a id="1385" href="Categories.Morphism.Properties.html#1212" class="Bound">h</a> <a id="1387" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1389" href="Categories.Morphism.Properties.html#1196" class="Bound">f</a><a id="1390" class="Symbol">)</a> <a id="1392" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1394" href="Categories.Morphism.Properties.html#1204" class="Bound">g</a> <a id="1396" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1398" href="Categories.Morphism.Properties.html#1220" class="Bound">i</a> <a id="1400" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1403" href="Categories.Morphism.Reasoning.Core.html#7037" class="Function">cancelInner</a> <a id="1415" class="Symbol">(</a><a id="1416" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="1421" href="Categories.Morphism.Properties.html#1223" class="Bound">iso</a><a id="1424" class="Symbol">)</a> <a id="1426" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1432" href="Categories.Morphism.Properties.html#1212" class="Bound">h</a> <a id="1434" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1436" href="Categories.Morphism.Properties.html#1220" class="Bound">i</a> <a id="1448" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1451" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="1456" href="Categories.Morphism.Properties.html#1227" class="Bound">iso′</a> <a id="1461" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1467" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="1483" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="1487" class="Symbol">}</a>
<a id="1491" class="Keyword">where</a> <a id="1497" class="Keyword">open</a> <a id="1502" href="Categories.Morphism.html#1528" class="Module">Iso</a>
<a id="Iso-≈"></a><a id="1507" href="Categories.Morphism.Properties.html#1507" class="Function">Iso-≈</a> <a id="1513" class="Symbol">:</a> <a id="1515" href="Categories.Morphism.Properties.html#405" class="Generalizable">f</a> <a id="1517" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1519" href="Categories.Morphism.Properties.html#409" class="Generalizable">h</a> <a id="1521" class="Symbol">→</a> <a id="1523" href="Categories.Morphism.html#1528" class="Record">Iso</a> <a id="1527" href="Categories.Morphism.Properties.html#405" class="Generalizable">f</a> <a id="1529" href="Categories.Morphism.Properties.html#407" class="Generalizable">g</a> <a id="1531" class="Symbol">→</a> <a id="1533" href="Categories.Morphism.html#1528" class="Record">Iso</a> <a id="1537" href="Categories.Morphism.Properties.html#409" class="Generalizable">h</a> <a id="1539" href="Categories.Morphism.Properties.html#411" class="Generalizable">i</a> <a id="1541" class="Symbol">→</a> <a id="1543" href="Categories.Morphism.Properties.html#407" class="Generalizable">g</a> <a id="1545" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1547" href="Categories.Morphism.Properties.html#411" class="Generalizable">i</a>
<a id="1549" href="Categories.Morphism.Properties.html#1507" class="Function">Iso-≈</a> <a id="1555" class="Symbol">{</a><a id="1556" class="Argument">f</a> <a id="1558" class="Symbol">=</a> <a id="1560" href="Categories.Morphism.Properties.html#1560" class="Bound">f</a><a id="1561" class="Symbol">}</a> <a id="1563" class="Symbol">{</a><a id="1564" class="Argument">h</a> <a id="1566" class="Symbol">=</a> <a id="1568" href="Categories.Morphism.Properties.html#1568" class="Bound">h</a><a id="1569" class="Symbol">}</a> <a id="1571" class="Symbol">{</a><a id="1572" class="Argument">g</a> <a id="1574" class="Symbol">=</a> <a id="1576" href="Categories.Morphism.Properties.html#1576" class="Bound">g</a><a id="1577" class="Symbol">}</a> <a id="1579" class="Symbol">{</a><a id="1580" class="Argument">i</a> <a id="1582" class="Symbol">=</a> <a id="1584" href="Categories.Morphism.Properties.html#1584" class="Bound">i</a><a id="1585" class="Symbol">}</a> <a id="1587" href="Categories.Morphism.Properties.html#1587" class="Bound">eq</a> <a id="1590" href="Categories.Morphism.Properties.html#1590" class="Bound">iso</a> <a id="1594" href="Categories.Morphism.Properties.html#1594" class="Bound">iso′</a> <a id="1599" class="Symbol">=</a> <a id="1601" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="1609" href="Categories.Morphism.Properties.html#1576" class="Bound">g</a> <a id="1621" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1624" href="Categories.Morphism.Reasoning.Core.html#3063" class="Function">introˡ</a> <a id="1631" class="Symbol">(</a><a id="1632" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="1637" href="Categories.Morphism.Properties.html#1594" class="Bound">iso′</a><a id="1641" class="Symbol">)</a> <a id="1643" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1647" class="Symbol">(</a><a id="1648" href="Categories.Morphism.Properties.html#1584" class="Bound">i</a> <a id="1650" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1652" href="Categories.Morphism.Properties.html#1568" class="Bound">h</a><a id="1653" class="Symbol">)</a> <a id="1655" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1657" href="Categories.Morphism.Properties.html#1576" class="Bound">g</a> <a id="1659" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="1663" class="Symbol">(</a><a id="1664" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="1672" href="Categories.Morphism.Properties.html#1587" class="Bound">eq</a><a id="1674" class="Symbol">)</a> <a id="1676" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="1684" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="1688" class="Symbol">(</a><a id="1689" href="Categories.Morphism.Properties.html#1584" class="Bound">i</a> <a id="1691" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1693" href="Categories.Morphism.Properties.html#1560" class="Bound">f</a><a id="1694" class="Symbol">)</a> <a id="1696" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1698" href="Categories.Morphism.Properties.html#1576" class="Bound">g</a> <a id="1700" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1703" href="Categories.Morphism.Reasoning.Core.html#6665" class="Function">cancelʳ</a> <a id="1711" class="Symbol">(</a><a id="1712" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="1717" href="Categories.Morphism.Properties.html#1590" class="Bound">iso</a><a id="1720" class="Symbol">)</a> <a id="1722" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1726" href="Categories.Morphism.Properties.html#1584" class="Bound">i</a> <a id="1738" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="1742" class="Keyword">where</a> <a id="1748" class="Keyword">open</a> <a id="1753" href="Categories.Morphism.html#1528" class="Module">Iso</a>
<a id="1758" class="Keyword">module</a> <a id="1765" href="Categories.Morphism.Properties.html#1765" class="Module">_</a> <a id="1767" class="Keyword">where</a>
<a id="1775" class="Keyword">open</a> <a id="1780" href="Categories.Morphism.html#1958" class="Module Operator">_≅_</a>
<a id="1787" href="Categories.Morphism.Properties.html#1787" class="Function">isos×≈⇒≈</a> <a id="1796" class="Symbol">:</a> <a id="1798" class="Symbol">∀</a> <a id="1800" class="Symbol">{</a><a id="1801" href="Categories.Morphism.Properties.html#1801" class="Bound">f</a> <a id="1803" href="Categories.Morphism.Properties.html#1803" class="Bound">g</a> <a id="1805" class="Symbol">:</a> <a id="1807" href="Categories.Morphism.Properties.html#387" class="Generalizable">A</a> <a id="1809" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="1811" href="Categories.Morphism.Properties.html#389" class="Generalizable">B</a><a id="1812" class="Symbol">}</a> <a id="1814" class="Symbol">→</a> <a id="1816" href="Categories.Morphism.Properties.html#409" class="Generalizable">h</a> <a id="1818" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1820" href="Categories.Morphism.Properties.html#411" class="Generalizable">i</a> <a id="1822" class="Symbol">→</a> <a id="1824" class="Symbol">(</a><a id="1825" href="Categories.Morphism.Properties.html#1825" class="Bound">iso₁</a> <a id="1830" class="Symbol">:</a> <a id="1832" href="Categories.Morphism.Properties.html#387" class="Generalizable">A</a> <a id="1834" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="1836" href="Categories.Morphism.Properties.html#391" class="Generalizable">C</a><a id="1837" class="Symbol">)</a> <a id="1839" class="Symbol">→</a> <a id="1841" class="Symbol">(</a><a id="1842" href="Categories.Morphism.Properties.html#1842" class="Bound">iso₂</a> <a id="1847" class="Symbol">:</a> <a id="1849" href="Categories.Morphism.Properties.html#389" class="Generalizable">B</a> <a id="1851" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="1853" href="Categories.Morphism.Properties.html#393" class="Generalizable">D</a><a id="1854" class="Symbol">)</a> <a id="1856" class="Symbol">→</a>
<a id="1873" href="Categories.Category.html#884" class="Function">CommutativeSquare</a> <a id="1891" href="Categories.Morphism.Properties.html#1801" class="Bound">f</a> <a id="1893" class="Symbol">(</a><a id="1894" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="1899" href="Categories.Morphism.Properties.html#1825" class="Bound">iso₁</a><a id="1903" class="Symbol">)</a> <a id="1905" class="Symbol">(</a><a id="1906" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="1911" href="Categories.Morphism.Properties.html#1842" class="Bound">iso₂</a><a id="1915" class="Symbol">)</a> <a id="1917" href="Categories.Morphism.Properties.html#409" class="Generalizable">h</a> <a id="1919" class="Symbol">→</a>
<a id="1936" href="Categories.Category.html#884" class="Function">CommutativeSquare</a> <a id="1954" href="Categories.Morphism.Properties.html#1803" class="Bound">g</a> <a id="1956" class="Symbol">(</a><a id="1957" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="1962" href="Categories.Morphism.Properties.html#1825" class="Bound">iso₁</a><a id="1966" class="Symbol">)</a> <a id="1968" class="Symbol">(</a><a id="1969" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="1974" href="Categories.Morphism.Properties.html#1842" class="Bound">iso₂</a><a id="1978" class="Symbol">)</a> <a id="1980" href="Categories.Morphism.Properties.html#411" class="Generalizable">i</a> <a id="1982" class="Symbol">→</a>
<a id="1999" href="Categories.Morphism.Properties.html#1801" class="Bound">f</a> <a id="2001" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="2003" href="Categories.Morphism.Properties.html#1803" class="Bound">g</a>
<a id="2007" href="Categories.Morphism.Properties.html#1787" class="Function">isos×≈⇒≈</a> <a id="2016" class="Symbol">{</a><a id="2017" class="Argument">h</a> <a id="2019" class="Symbol">=</a> <a id="2021" href="Categories.Morphism.Properties.html#2021" class="Bound">h</a><a id="2022" class="Symbol">}</a> <a id="2024" class="Symbol">{</a><a id="2025" class="Argument">i</a> <a id="2027" class="Symbol">=</a> <a id="2029" href="Categories.Morphism.Properties.html#2029" class="Bound">i</a><a id="2030" class="Symbol">}</a> <a id="2032" class="Symbol">{</a><a id="2033" class="Argument">f</a> <a id="2035" class="Symbol">=</a> <a id="2037" href="Categories.Morphism.Properties.html#2037" class="Bound">f</a><a id="2038" class="Symbol">}</a> <a id="2040" class="Symbol">{</a><a id="2041" class="Argument">g</a> <a id="2043" class="Symbol">=</a> <a id="2045" href="Categories.Morphism.Properties.html#2045" class="Bound">g</a><a id="2046" class="Symbol">}</a> <a id="2048" href="Categories.Morphism.Properties.html#2048" class="Bound">eq</a> <a id="2051" href="Categories.Morphism.Properties.html#2051" class="Bound">iso₁</a> <a id="2056" href="Categories.Morphism.Properties.html#2056" class="Bound">iso₂</a> <a id="2061" href="Categories.Morphism.Properties.html#2061" class="Bound">sq₁</a> <a id="2065" href="Categories.Morphism.Properties.html#2065" class="Bound">sq₂</a> <a id="2069" class="Symbol">=</a> <a id="2071" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="2081" href="Categories.Morphism.Properties.html#2037" class="Bound">f</a> <a id="2083" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2086" href="Categories.Morphism.Reasoning.Iso.html#720" class="Function">switch-fromtoˡ</a> <a id="2101" href="Categories.Morphism.Properties.html#2056" class="Bound">iso₂</a> <a id="2106" href="Categories.Morphism.Properties.html#2061" class="Bound">sq₁</a> <a id="2110" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2116" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="2119" href="Categories.Morphism.Properties.html#2056" class="Bound">iso₂</a> <a id="2124" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2126" href="Categories.Morphism.Properties.html#2021" class="Bound">h</a> <a id="2128" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2130" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="2135" href="Categories.Morphism.Properties.html#2051" class="Bound">iso₁</a> <a id="2140" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2143" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="2151" class="Symbol">(</a><a id="2152" href="Categories.Morphism.Properties.html#2048" class="Bound">eq</a> <a id="2155" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="2163" class="Symbol">)</a> <a id="2165" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2171" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="2174" href="Categories.Morphism.Properties.html#2056" class="Bound">iso₂</a> <a id="2179" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2181" href="Categories.Morphism.Properties.html#2029" class="Bound">i</a> <a id="2183" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2185" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="2190" href="Categories.Morphism.Properties.html#2051" class="Bound">iso₁</a> <a id="2195" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="2199" href="Categories.Morphism.Reasoning.Iso.html#720" class="Function">switch-fromtoˡ</a> <a id="2214" href="Categories.Morphism.Properties.html#2056" class="Bound">iso₂</a> <a id="2219" href="Categories.Morphism.Properties.html#2065" class="Bound">sq₂</a> <a id="2223" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="2229" href="Categories.Morphism.Properties.html#2045" class="Bound">g</a> <a id="2231" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="id-is-iso"></a><a id="2234" href="Categories.Morphism.Properties.html#2234" class="Function">id-is-iso</a> <a id="2244" class="Symbol">:</a> <a id="2246" class="Symbol">∀</a> <a id="2248" class="Symbol">{</a><a id="2249" href="Categories.Morphism.Properties.html#2249" class="Bound">X</a><a id="2250" class="Symbol">}</a> <a id="2252" class="Symbol">→</a> <a id="2254" href="Categories.Morphism.html#1826" class="Record">IsIso</a> <a id="2260" class="Symbol">(</a><a id="2261" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="2264" class="Symbol">{</a><a id="2265" href="Categories.Morphism.Properties.html#2249" class="Bound">X</a><a id="2266" class="Symbol">})</a>
<a id="2269" href="Categories.Morphism.Properties.html#2234" class="Function">id-is-iso</a> <a id="2279" class="Symbol">=</a> <a id="2281" class="Keyword">record</a>
<a id="2290" class="Symbol">{</a> <a id="2292" href="Categories.Morphism.html#1879" class="Field">inv</a> <a id="2296" class="Symbol">=</a> <a id="2298" href="Categories.Category.Core.html#630" class="Field">id</a>
<a id="2303" class="Symbol">;</a> <a id="2305" href="Categories.Morphism.html#1895" class="Field">iso</a> <a id="2309" class="Symbol">=</a> <a id="2311" class="Keyword">record</a>
<a id="2322" class="Symbol">{</a> <a id="2324" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="2329" class="Symbol">=</a> <a id="2331" href="Categories.Category.Core.html#1339" class="Field">identity²</a>
<a id="2345" class="Symbol">;</a> <a id="2347" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="2352" class="Symbol">=</a> <a id="2354" href="Categories.Category.Core.html#1339" class="Field">identity²</a>
<a id="2368" class="Symbol">}</a>
<a id="2372" class="Symbol">}</a>
<a id="2379" class="Comment">--------------------------------------------------------------------------------</a>
<a id="2460" class="Comment">-- Monomorphisms</a>
<a id="Mono-∘₂"></a><a id="2478" href="Categories.Morphism.Properties.html#2478" class="Function">Mono-∘₂</a> <a id="2486" class="Symbol">:</a> <a id="2488" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="2493" class="Symbol">(</a><a id="2494" href="Categories.Morphism.Properties.html#405" class="Generalizable">f</a> <a id="2496" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2498" href="Categories.Morphism.Properties.html#407" class="Generalizable">g</a><a id="2499" class="Symbol">)</a> <a id="2501" class="Symbol">→</a> <a id="2503" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="2508" href="Categories.Morphism.Properties.html#407" class="Generalizable">g</a>
<a id="2510" href="Categories.Morphism.Properties.html#2478" class="Function">Mono-∘₂</a> <a id="2518" class="Symbol">{</a><a id="2519" class="Argument">f</a> <a id="2521" class="Symbol">=</a> <a id="2523" href="Categories.Morphism.Properties.html#2523" class="Bound">f</a><a id="2524" class="Symbol">}</a> <a id="2526" class="Symbol">{</a><a id="2527" class="Argument">g</a> <a id="2529" class="Symbol">=</a> <a id="2531" href="Categories.Morphism.Properties.html#2531" class="Bound">g</a><a id="2532" class="Symbol">}</a> <a id="2534" href="Categories.Morphism.Properties.html#2534" class="Bound">fg-mono</a> <a id="2542" href="Categories.Morphism.Properties.html#2542" class="Bound">g₁</a> <a id="2545" href="Categories.Morphism.Properties.html#2545" class="Bound">g₂</a> <a id="2548" href="Categories.Morphism.Properties.html#2548" class="Bound">eq</a> <a id="2551" class="Symbol">=</a> <a id="2553" href="Categories.Morphism.Properties.html#2534" class="Bound">fg-mono</a> <a id="2561" href="Categories.Morphism.Properties.html#2542" class="Bound">g₁</a> <a id="2564" href="Categories.Morphism.Properties.html#2545" class="Bound">g₂</a> <a id="2567" class="Symbol">(</a><a id="2568" href="Categories.Morphism.Reasoning.Core.html#3392" class="Function">extendˡ</a> <a id="2576" href="Categories.Morphism.Properties.html#2548" class="Bound">eq</a><a id="2578" class="Symbol">)</a>
<a id="2581" class="Comment">-- This might be trivial, but it also shouldn&#39;t be proved more than once!</a>
<a id="Mono-id"></a><a id="2655" href="Categories.Morphism.Properties.html#2655" class="Function">Mono-id</a> <a id="2663" class="Symbol">:</a> <a id="2665" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="2670" class="Symbol">{</a><a id="2671" class="Argument">A</a> <a id="2673" class="Symbol">=</a> <a id="2675" href="Categories.Morphism.Properties.html#387" class="Generalizable">A</a><a id="2676" class="Symbol">}</a> <a id="2678" href="Categories.Category.Core.html#630" class="Field">id</a>
<a id="2681" href="Categories.Morphism.Properties.html#2655" class="Function">Mono-id</a> <a id="2689" href="Categories.Morphism.Properties.html#2689" class="Bound">g₁</a> <a id="2692" href="Categories.Morphism.Properties.html#2692" class="Bound">g₂</a> <a id="2695" href="Categories.Morphism.Properties.html#2695" class="Bound">eq</a> <a id="2698" class="Symbol">=</a> <a id="2700" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="2708" href="Categories.Morphism.Properties.html#2689" class="Bound">g₁</a> <a id="2716" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="2720" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="2730" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="2734" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="2737" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2739" href="Categories.Morphism.Properties.html#2689" class="Bound">g₁</a> <a id="2742" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2745" href="Categories.Morphism.Properties.html#2695" class="Bound">eq</a> <a id="2748" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2752" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="2755" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2757" href="Categories.Morphism.Properties.html#2692" class="Bound">g₂</a> <a id="2760" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2763" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="2773" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2777" href="Categories.Morphism.Properties.html#2692" class="Bound">g₂</a> <a id="2780" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="Mono-∘"></a><a id="2783" href="Categories.Morphism.Properties.html#2783" class="Function">Mono-∘</a> <a id="2790" class="Symbol">:</a> <a id="2792" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="2797" href="Categories.Morphism.Properties.html#405" class="Generalizable">f</a> <a id="2799" class="Symbol">→</a> <a id="2801" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="2806" href="Categories.Morphism.Properties.html#407" class="Generalizable">g</a> <a id="2808" class="Symbol">→</a> <a id="2810" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="2815" class="Symbol">(</a><a id="2816" href="Categories.Morphism.Properties.html#405" class="Generalizable">f</a> <a id="2818" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2820" href="Categories.Morphism.Properties.html#407" class="Generalizable">g</a><a id="2821" class="Symbol">)</a>
<a id="2823" href="Categories.Morphism.Properties.html#2783" class="Function">Mono-∘</a> <a id="2830" class="Symbol">{</a><a id="2831" class="Argument">f</a> <a id="2833" class="Symbol">=</a> <a id="2835" href="Categories.Morphism.Properties.html#2835" class="Bound">f</a><a id="2836" class="Symbol">}</a> <a id="2838" class="Symbol">{</a><a id="2839" class="Argument">g</a> <a id="2841" class="Symbol">=</a> <a id="2843" href="Categories.Morphism.Properties.html#2843" class="Bound">g</a><a id="2844" class="Symbol">}</a> <a id="2846" href="Categories.Morphism.Properties.html#2846" class="Bound">f-mono</a> <a id="2853" href="Categories.Morphism.Properties.html#2853" class="Bound">g-mono</a> <a id="2860" href="Categories.Morphism.Properties.html#2860" class="Bound">g₁</a> <a id="2863" href="Categories.Morphism.Properties.html#2863" class="Bound">g₂</a> <a id="2866" href="Categories.Morphism.Properties.html#2866" class="Bound">eq</a> <a id="2869" class="Symbol">=</a>
<a id="2873" href="Categories.Morphism.Properties.html#2853" class="Bound">g-mono</a> <a id="2880" href="Categories.Morphism.Properties.html#2860" class="Bound">g₁</a> <a id="2883" href="Categories.Morphism.Properties.html#2863" class="Bound">g₂</a> <a id="2886" class="Symbol">(</a><a id="2887" href="Categories.Morphism.Properties.html#2846" class="Bound">f-mono</a> <a id="2894" class="Symbol">(</a><a id="2895" href="Categories.Morphism.Properties.html#2843" class="Bound">g</a> <a id="2897" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2899" href="Categories.Morphism.Properties.html#2860" class="Bound">g₁</a><a id="2901" class="Symbol">)</a> <a id="2903" class="Symbol">(</a><a id="2904" href="Categories.Morphism.Properties.html#2843" class="Bound">g</a> <a id="2906" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2908" href="Categories.Morphism.Properties.html#2863" class="Bound">g₂</a><a id="2910" class="Symbol">)</a> <a id="2912" class="Symbol">(</a><a id="2913" href="Categories.Category.Core.html#1004" class="Field">sym-assoc</a> <a id="2923" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2925" href="Categories.Morphism.Properties.html#2866" class="Bound">eq</a> <a id="2928" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2930" href="Categories.Category.Core.html#715" class="Field">assoc</a><a id="2935" class="Symbol">))</a>
<a id="id↣"></a><a id="2939" href="Categories.Morphism.Properties.html#2939" class="Function">id↣</a> <a id="2943" class="Symbol">:</a> <a id="2945" class="Symbol">∀</a> <a id="2947" class="Symbol">{</a><a id="2948" href="Categories.Morphism.Properties.html#2948" class="Bound">A</a><a id="2949" class="Symbol">}</a> <a id="2951" class="Symbol">→</a> <a id="2953" href="Categories.Morphism.Properties.html#2948" class="Bound">A</a> <a id="2955" href="Categories.Morphism.html#789" class="Record Operator">↣</a> <a id="2957" href="Categories.Morphism.Properties.html#2948" class="Bound">A</a>
<a id="2959" href="Categories.Morphism.Properties.html#2939" class="Function">id↣</a> <a id="2963" class="Symbol">=</a> <a id="2965" class="Keyword">record</a> <a id="2972" class="Symbol">{</a> <a id="2974" href="Categories.Morphism.html#841" class="Field">mor</a> <a id="2978" class="Symbol">=</a> <a id="2980" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="2983" class="Symbol">;</a> <a id="2985" href="Categories.Morphism.html#858" class="Field">mono</a> <a id="2990" class="Symbol">=</a> <a id="2992" href="Categories.Morphism.Properties.html#2655" class="Function">Mono-id</a> <a id="3000" class="Symbol">}</a>
<a id="3003" class="Keyword">infixr</a> <a id="3010" class="Number">9</a> <a id="3012" href="Categories.Morphism.Properties.html#3017" class="Function Operator">_∘↣_</a>
<a id="_∘↣_"></a><a id="3017" href="Categories.Morphism.Properties.html#3017" class="Function Operator">_∘↣_</a> <a id="3022" class="Symbol">:</a> <a id="3024" href="Categories.Morphism.Properties.html#389" class="Generalizable">B</a> <a id="3026" href="Categories.Morphism.html#789" class="Record Operator">↣</a> <a id="3028" href="Categories.Morphism.Properties.html#391" class="Generalizable">C</a> <a id="3030" class="Symbol">→</a> <a id="3032" href="Categories.Morphism.Properties.html#387" class="Generalizable">A</a> <a id="3034" href="Categories.Morphism.html#789" class="Record Operator">↣</a> <a id="3036" href="Categories.Morphism.Properties.html#389" class="Generalizable">B</a> <a id="3038" class="Symbol">→</a> <a id="3040" href="Categories.Morphism.Properties.html#387" class="Generalizable">A</a> <a id="3042" href="Categories.Morphism.html#789" class="Record Operator">↣</a> <a id="3044" href="Categories.Morphism.Properties.html#391" class="Generalizable">C</a>
<a id="3046" href="Categories.Morphism.Properties.html#3046" class="Bound">f</a> <a id="3048" href="Categories.Morphism.Properties.html#3017" class="Function Operator">∘↣</a> <a id="3051" href="Categories.Morphism.Properties.html#3051" class="Bound">g</a> <a id="3053" class="Symbol">=</a> <a id="3055" class="Keyword">record</a> <a id="3062" class="Symbol">{</a> <a id="3064" href="Categories.Morphism.html#841" class="Field">mor</a> <a id="3068" class="Symbol">=</a> <a id="3070" href="Categories.Morphism.html#841" class="Field">mor</a> <a id="3074" href="Categories.Morphism.Properties.html#3046" class="Bound">f</a> <a id="3076" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3078" href="Categories.Morphism.html#841" class="Field">mor</a> <a id="3082" href="Categories.Morphism.Properties.html#3051" class="Bound">g</a> <a id="3084" class="Symbol">;</a> <a id="3086" href="Categories.Morphism.html#858" class="Field">mono</a> <a id="3091" class="Symbol">=</a> <a id="3093" href="Categories.Morphism.Properties.html#2783" class="Function">Mono-∘</a> <a id="3100" class="Symbol">(</a><a id="3101" href="Categories.Morphism.html#858" class="Field">mono</a> <a id="3106" href="Categories.Morphism.Properties.html#3046" class="Bound">f</a><a id="3107" class="Symbol">)</a> <a id="3109" class="Symbol">(</a><a id="3110" href="Categories.Morphism.html#858" class="Field">mono</a> <a id="3115" href="Categories.Morphism.Properties.html#3051" class="Bound">g</a><a id="3116" class="Symbol">)</a> <a id="3118" class="Symbol">}</a>
<a id="3122" class="Keyword">where</a>
<a id="3132" class="Keyword">open</a> <a id="3137" href="Categories.Morphism.html#789" class="Module Operator">_↣_</a>
<a id="3141" class="Comment">--------------------------------------------------------------------------------</a>
<a id="3222" class="Comment">-- Epimorphisms</a>
<a id="Epi-∘₂"></a><a id="3239" href="Categories.Morphism.Properties.html#3239" class="Function">Epi-∘₂</a> <a id="3246" class="Symbol">:</a> <a id="3248" href="Categories.Morphism.html#875" class="Function">Epi</a> <a id="3252" class="Symbol">(</a><a id="3253" href="Categories.Morphism.Properties.html#405" class="Generalizable">f</a> <a id="3255" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3257" href="Categories.Morphism.Properties.html#407" class="Generalizable">g</a><a id="3258" class="Symbol">)</a> <a id="3260" class="Symbol">→</a> <a id="3262" href="Categories.Morphism.html#875" class="Function">Epi</a> <a id="3266" href="Categories.Morphism.Properties.html#405" class="Generalizable">f</a>
<a id="3268" href="Categories.Morphism.Properties.html#3239" class="Function">Epi-∘₂</a> <a id="3275" class="Symbol">{</a><a id="3276" class="Argument">f</a> <a id="3278" class="Symbol">=</a> <a id="3280" href="Categories.Morphism.Properties.html#3280" class="Bound">f</a><a id="3281" class="Symbol">}</a> <a id="3283" class="Symbol">{</a><a id="3284" class="Argument">g</a> <a id="3286" class="Symbol">=</a> <a id="3288" href="Categories.Morphism.Properties.html#3288" class="Bound">g</a><a id="3289" class="Symbol">}</a> <a id="3291" href="Categories.Morphism.Properties.html#3291" class="Bound">fg-epi</a> <a id="3298" href="Categories.Morphism.Properties.html#3298" class="Bound">g₁</a> <a id="3301" href="Categories.Morphism.Properties.html#3301" class="Bound">g₂</a> <a id="3304" href="Categories.Morphism.Properties.html#3304" class="Bound">eq</a> <a id="3307" class="Symbol">=</a> <a id="3309" href="Categories.Morphism.Properties.html#3291" class="Bound">fg-epi</a> <a id="3316" href="Categories.Morphism.Properties.html#3298" class="Bound">g₁</a> <a id="3319" href="Categories.Morphism.Properties.html#3301" class="Bound">g₂</a> <a id="3322" class="Symbol">(</a><a id="3323" href="Categories.Morphism.Reasoning.Core.html#3587" class="Function">extendʳ</a> <a id="3331" href="Categories.Morphism.Properties.html#3304" class="Bound">eq</a><a id="3333" class="Symbol">)</a>
<a id="Epi-id"></a><a id="3336" href="Categories.Morphism.Properties.html#3336" class="Function">Epi-id</a> <a id="3343" class="Symbol">:</a> <a id="3345" href="Categories.Morphism.html#875" class="Function">Epi</a> <a id="3349" class="Symbol">{</a><a id="3350" class="Argument">A</a> <a id="3352" class="Symbol">=</a> <a id="3354" href="Categories.Morphism.Properties.html#387" class="Generalizable">A</a><a id="3355" class="Symbol">}</a> <a id="3357" href="Categories.Category.Core.html#630" class="Field">id</a>
<a id="3360" href="Categories.Morphism.Properties.html#3336" class="Function">Epi-id</a> <a id="3367" href="Categories.Morphism.Properties.html#3367" class="Bound">g₁</a> <a id="3370" href="Categories.Morphism.Properties.html#3370" class="Bound">g₂</a> <a id="3373" href="Categories.Morphism.Properties.html#3373" class="Bound">eq</a> <a id="3376" class="Symbol">=</a> <a id="3378" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="3386" href="Categories.Morphism.Properties.html#3367" class="Bound">g₁</a> <a id="3394" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="3398" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a> <a id="3408" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="3412" href="Categories.Morphism.Properties.html#3367" class="Bound">g₁</a> <a id="3415" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3417" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="3420" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="3423" href="Categories.Morphism.Properties.html#3373" class="Bound">eq</a> <a id="3426" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="3430" href="Categories.Morphism.Properties.html#3370" class="Bound">g₂</a> <a id="3433" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3435" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="3438" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="3441" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a> <a id="3451" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="3455" href="Categories.Morphism.Properties.html#3370" class="Bound">g₂</a> <a id="3458" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="Epi-∘"></a><a id="3461" href="Categories.Morphism.Properties.html#3461" class="Function">Epi-∘</a> <a id="3467" class="Symbol">:</a> <a id="3469" href="Categories.Morphism.html#875" class="Function">Epi</a> <a id="3473" href="Categories.Morphism.Properties.html#405" class="Generalizable">f</a> <a id="3475" class="Symbol">→</a> <a id="3477" href="Categories.Morphism.html#875" class="Function">Epi</a> <a id="3481" href="Categories.Morphism.Properties.html#407" class="Generalizable">g</a> <a id="3483" class="Symbol">→</a> <a id="3485" href="Categories.Morphism.html#875" class="Function">Epi</a> <a id="3489" class="Symbol">(</a><a id="3490" href="Categories.Morphism.Properties.html#405" class="Generalizable">f</a> <a id="3492" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3494" href="Categories.Morphism.Properties.html#407" class="Generalizable">g</a><a id="3495" class="Symbol">)</a>
<a id="3497" href="Categories.Morphism.Properties.html#3461" class="Function">Epi-∘</a> <a id="3503" class="Symbol">{</a><a id="3504" class="Argument">f</a> <a id="3506" class="Symbol">=</a> <a id="3508" href="Categories.Morphism.Properties.html#3508" class="Bound">f</a><a id="3509" class="Symbol">}</a> <a id="3511" class="Symbol">{</a><a id="3512" class="Argument">g</a> <a id="3514" class="Symbol">=</a> <a id="3516" href="Categories.Morphism.Properties.html#3516" class="Bound">g</a><a id="3517" class="Symbol">}</a> <a id="3519" href="Categories.Morphism.Properties.html#3519" class="Bound">f-epi</a> <a id="3525" href="Categories.Morphism.Properties.html#3525" class="Bound">g-epi</a> <a id="3531" href="Categories.Morphism.Properties.html#3531" class="Bound">g₁</a> <a id="3534" href="Categories.Morphism.Properties.html#3534" class="Bound">g₂</a> <a id="3537" href="Categories.Morphism.Properties.html#3537" class="Bound">eq</a> <a id="3540" class="Symbol">=</a>
<a id="3544" href="Categories.Morphism.Properties.html#3519" class="Bound">f-epi</a> <a id="3550" href="Categories.Morphism.Properties.html#3531" class="Bound">g₁</a> <a id="3553" href="Categories.Morphism.Properties.html#3534" class="Bound">g₂</a> <a id="3556" class="Symbol">(</a><a id="3557" href="Categories.Morphism.Properties.html#3525" class="Bound">g-epi</a> <a id="3563" class="Symbol">(</a><a id="3564" href="Categories.Morphism.Properties.html#3531" class="Bound">g₁</a> <a id="3567" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3569" href="Categories.Morphism.Properties.html#3508" class="Bound">f</a><a id="3570" class="Symbol">)</a> <a id="3572" class="Symbol">(</a><a id="3573" href="Categories.Morphism.Properties.html#3534" class="Bound">g₂</a> <a id="3576" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3578" href="Categories.Morphism.Properties.html#3508" class="Bound">f</a><a id="3579" class="Symbol">)</a> <a id="3581" class="Symbol">(</a><a id="3582" href="Categories.Category.Core.html#715" class="Field">assoc</a> <a id="3588" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="3590" href="Categories.Morphism.Properties.html#3537" class="Bound">eq</a> <a id="3593" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="3595" href="Categories.Category.Core.html#1004" class="Field">sym-assoc</a><a id="3604" class="Symbol">))</a>
<a id="id↠"></a><a id="3608" href="Categories.Morphism.Properties.html#3608" class="Function">id↠</a> <a id="3612" class="Symbol">:</a> <a id="3614" class="Symbol">∀</a> <a id="3616" class="Symbol">{</a><a id="3617" href="Categories.Morphism.Properties.html#3617" class="Bound">A</a><a id="3618" class="Symbol">}</a> <a id="3620" class="Symbol">→</a> <a id="3622" href="Categories.Morphism.Properties.html#3617" class="Bound">A</a> <a id="3624" href="Categories.Morphism.html#1154" class="Record Operator">↠</a> <a id="3626" href="Categories.Morphism.Properties.html#3617" class="Bound">A</a>
<a id="3628" href="Categories.Morphism.Properties.html#3608" class="Function">id↠</a> <a id="3632" class="Symbol">=</a> <a id="3634" class="Keyword">record</a> <a id="3641" class="Symbol">{</a> <a id="3643" href="Categories.Morphism.html#1206" class="Field">mor</a> <a id="3647" class="Symbol">=</a> <a id="3649" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="3652" class="Symbol">;</a> <a id="3654" href="Categories.Morphism.html#1222" class="Field">epi</a> <a id="3658" class="Symbol">=</a> <a id="3660" href="Categories.Morphism.Properties.html#3336" class="Function">Epi-id</a> <a id="3667" class="Symbol">}</a>
<a id="3670" class="Keyword">infixr</a> <a id="3677" class="Number">9</a> <a id="3679" href="Categories.Morphism.Properties.html#3685" class="Function Operator">_∘↠_</a>
<a id="_∘↠_"></a><a id="3685" href="Categories.Morphism.Properties.html#3685" class="Function Operator">_∘↠_</a> <a id="3690" class="Symbol">:</a> <a id="3692" href="Categories.Morphism.Properties.html#389" class="Generalizable">B</a> <a id="3694" href="Categories.Morphism.html#1154" class="Record Operator">↠</a> <a id="3696" href="Categories.Morphism.Properties.html#391" class="Generalizable">C</a> <a id="3698" class="Symbol">→</a> <a id="3700" href="Categories.Morphism.Properties.html#387" class="Generalizable">A</a> <a id="3702" href="Categories.Morphism.html#1154" class="Record Operator">↠</a> <a id="3704" href="Categories.Morphism.Properties.html#389" class="Generalizable">B</a> <a id="3706" class="Symbol">→</a> <a id="3708" href="Categories.Morphism.Properties.html#387" class="Generalizable">A</a> <a id="3710" href="Categories.Morphism.html#1154" class="Record Operator">↠</a> <a id="3712" href="Categories.Morphism.Properties.html#391" class="Generalizable">C</a>
<a id="3714" href="Categories.Morphism.Properties.html#3714" class="Bound">f</a> <a id="3716" href="Categories.Morphism.Properties.html#3685" class="Function Operator">∘↠</a> <a id="3719" href="Categories.Morphism.Properties.html#3719" class="Bound">g</a> <a id="3721" class="Symbol">=</a> <a id="3723" class="Keyword">record</a> <a id="3730" class="Symbol">{</a> <a id="3732" href="Categories.Morphism.html#1206" class="Field">mor</a> <a id="3736" class="Symbol">=</a> <a id="3738" href="Categories.Morphism.html#1206" class="Field">mor</a> <a id="3742" href="Categories.Morphism.Properties.html#3714" class="Bound">f</a> <a id="3744" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3746" href="Categories.Morphism.html#1206" class="Field">mor</a> <a id="3750" href="Categories.Morphism.Properties.html#3719" class="Bound">g</a> <a id="3752" class="Symbol">;</a> <a id="3754" href="Categories.Morphism.html#1222" class="Field">epi</a> <a id="3758" class="Symbol">=</a> <a id="3760" href="Categories.Morphism.Properties.html#3461" class="Function">Epi-∘</a> <a id="3766" class="Symbol">(</a><a id="3767" href="Categories.Morphism.html#1222" class="Field">epi</a> <a id="3771" href="Categories.Morphism.Properties.html#3714" class="Bound">f</a><a id="3772" class="Symbol">)</a> <a id="3774" class="Symbol">(</a><a id="3775" href="Categories.Morphism.html#1222" class="Field">epi</a> <a id="3779" href="Categories.Morphism.Properties.html#3719" class="Bound">g</a><a id="3780" class="Symbol">)</a> <a id="3782" class="Symbol">}</a>
<a id="3786" class="Keyword">where</a>
<a id="3796" class="Keyword">open</a> <a id="3801" href="Categories.Morphism.html#1154" class="Module Operator">_↠_</a>
</pre></body></html>
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