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48 lines
21 KiB
HTML
48 lines
21 KiB
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<html><head><meta charset="utf-8"><title>Categories.Diagram.Pushout</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="37" class="Keyword">open</a> <a id="42" class="Keyword">import</a> <a id="49" href="Categories.Category.Core.html" class="Module">Categories.Category.Core</a> <a id="74" class="Keyword">using</a> <a id="80" class="Symbol">(</a><a id="81" href="Categories.Category.Core.html#442" class="Record">Category</a><a id="89" class="Symbol">)</a>
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<a id="92" class="Keyword">module</a> <a id="99" href="Categories.Diagram.Pushout.html" class="Module">Categories.Diagram.Pushout</a> <a id="126" class="Symbol">{</a><a id="127" href="Categories.Diagram.Pushout.html#127" class="Bound">o</a> <a id="129" href="Categories.Diagram.Pushout.html#129" class="Bound">ℓ</a> <a id="131" href="Categories.Diagram.Pushout.html#131" class="Bound">e</a><a id="132" class="Symbol">}</a> <a id="134" class="Symbol">(</a><a id="135" href="Categories.Diagram.Pushout.html#135" class="Bound">C</a> <a id="137" class="Symbol">:</a> <a id="139" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="148" href="Categories.Diagram.Pushout.html#127" class="Bound">o</a> <a id="150" href="Categories.Diagram.Pushout.html#129" class="Bound">ℓ</a> <a id="152" href="Categories.Diagram.Pushout.html#131" class="Bound">e</a><a id="153" class="Symbol">)</a> <a id="155" class="Keyword">where</a>
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<a id="162" class="Keyword">open</a> <a id="167" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="176" href="Categories.Diagram.Pushout.html#135" class="Bound">C</a>
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<a id="178" class="Keyword">open</a> <a id="183" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
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<a id="196" class="Keyword">open</a> <a id="201" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
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<a id="208" class="Keyword">open</a> <a id="213" class="Keyword">import</a> <a id="220" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="250" href="Categories.Diagram.Pushout.html#135" class="Bound">C</a> as <a id="Square"></a><a id="255" href="Categories.Diagram.Pushout.html#255" class="Module">Square</a>
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<a id="264" class="Keyword">renaming</a> <a id="273" class="Symbol">(</a><a id="274" href="Categories.Morphism.Reasoning.Core.html#4290" class="Function">glue</a> <a id="279" class="Symbol">to</a> <a id="282" class="Function">glue-square</a><a id="293" class="Symbol">)</a> <a id="295" class="Keyword">hiding</a> <a id="302" class="Symbol">(</a><a id="303" href="Categories.Morphism.Reasoning.Core.html#1075" class="Function">id-unique</a><a id="312" class="Symbol">)</a>
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<a id="315" class="Keyword">open</a> <a id="320" class="Keyword">import</a> <a id="327" href="Level.html" class="Module">Level</a>
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<a id="334" class="Keyword">private</a>
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<a id="344" class="Keyword">variable</a>
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<a id="357" href="Categories.Diagram.Pushout.html#357" class="Generalizable">A</a> <a id="359" href="Categories.Diagram.Pushout.html#359" class="Generalizable">B</a> <a id="361" href="Categories.Diagram.Pushout.html#361" class="Generalizable">E</a> <a id="363" href="Categories.Diagram.Pushout.html#363" class="Generalizable">X</a> <a id="365" href="Categories.Diagram.Pushout.html#365" class="Generalizable">Y</a> <a id="367" href="Categories.Diagram.Pushout.html#367" class="Generalizable">Z</a> <a id="369" class="Symbol">:</a> <a id="371" href="Categories.Category.Core.html#559" class="Field">Obj</a>
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<a id="379" href="Categories.Diagram.Pushout.html#379" class="Generalizable">f</a> <a id="381" href="Categories.Diagram.Pushout.html#381" class="Generalizable">g</a> <a id="383" href="Categories.Diagram.Pushout.html#383" class="Generalizable">h</a> <a id="385" href="Categories.Diagram.Pushout.html#385" class="Generalizable">j</a> <a id="387" class="Symbol">:</a> <a id="389" href="Categories.Diagram.Pushout.html#357" class="Generalizable">A</a> <a id="391" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="393" href="Categories.Diagram.Pushout.html#359" class="Generalizable">B</a>
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<a id="396" class="Keyword">record</a> <a id="Pushout"></a><a id="403" href="Categories.Diagram.Pushout.html#403" class="Record">Pushout</a> <a id="411" class="Symbol">(</a><a id="412" href="Categories.Diagram.Pushout.html#412" class="Bound">f</a> <a id="414" class="Symbol">:</a> <a id="416" href="Categories.Diagram.Pushout.html#363" class="Generalizable">X</a> <a id="418" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="420" href="Categories.Diagram.Pushout.html#365" class="Generalizable">Y</a><a id="421" class="Symbol">)</a> <a id="423" class="Symbol">(</a><a id="424" href="Categories.Diagram.Pushout.html#424" class="Bound">g</a> <a id="426" class="Symbol">:</a> <a id="428" href="Categories.Diagram.Pushout.html#363" class="Generalizable">X</a> <a id="430" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="432" href="Categories.Diagram.Pushout.html#367" class="Generalizable">Z</a><a id="433" class="Symbol">)</a> <a id="435" class="Symbol">:</a> <a id="437" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="441" class="Symbol">(</a><a id="442" href="Categories.Diagram.Pushout.html#127" class="Bound">o</a> <a id="444" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="446" href="Categories.Diagram.Pushout.html#129" class="Bound">ℓ</a> <a id="448" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="450" href="Categories.Diagram.Pushout.html#131" class="Bound">e</a><a id="451" class="Symbol">)</a> <a id="453" class="Keyword">where</a>
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<a id="461" class="Keyword">field</a>
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<a id="471" class="Symbol">{</a><a id="Pushout.Q"></a><a id="472" href="Categories.Diagram.Pushout.html#472" class="Field">Q</a><a id="473" class="Symbol">}</a> <a id="475" class="Symbol">:</a> <a id="477" href="Categories.Category.Core.html#559" class="Field">Obj</a>
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<a id="Pushout.i₁"></a><a id="485" href="Categories.Diagram.Pushout.html#485" class="Field">i₁</a> <a id="489" class="Symbol">:</a> <a id="491" href="Categories.Diagram.Pushout.html#420" class="Bound">Y</a> <a id="493" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="495" href="Categories.Diagram.Pushout.html#472" class="Field">Q</a>
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<a id="Pushout.i₂"></a><a id="501" href="Categories.Diagram.Pushout.html#501" class="Field">i₂</a> <a id="505" class="Symbol">:</a> <a id="507" href="Categories.Diagram.Pushout.html#432" class="Bound">Z</a> <a id="509" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="511" href="Categories.Diagram.Pushout.html#472" class="Field">Q</a>
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<a id="516" class="Keyword">field</a>
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<a id="Pushout.commute"></a><a id="526" href="Categories.Diagram.Pushout.html#526" class="Field">commute</a> <a id="536" class="Symbol">:</a> <a id="538" href="Categories.Diagram.Pushout.html#485" class="Field">i₁</a> <a id="541" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="543" href="Categories.Diagram.Pushout.html#412" class="Bound">f</a> <a id="545" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="547" href="Categories.Diagram.Pushout.html#501" class="Field">i₂</a> <a id="550" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="552" href="Categories.Diagram.Pushout.html#424" class="Bound">g</a>
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<a id="Pushout.universal"></a><a id="558" href="Categories.Diagram.Pushout.html#558" class="Field">universal</a> <a id="568" class="Symbol">:</a> <a id="570" class="Symbol">{</a><a id="571" href="Categories.Diagram.Pushout.html#571" class="Bound">h₁</a> <a id="574" class="Symbol">:</a> <a id="576" href="Categories.Diagram.Pushout.html#420" class="Bound">Y</a> <a id="578" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="580" href="Categories.Diagram.Pushout.html#359" class="Generalizable">B</a><a id="581" class="Symbol">}</a> <a id="583" class="Symbol">{</a><a id="584" href="Categories.Diagram.Pushout.html#584" class="Bound">h₂</a> <a id="587" class="Symbol">:</a> <a id="589" href="Categories.Diagram.Pushout.html#432" class="Bound">Z</a> <a id="591" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="593" href="Categories.Diagram.Pushout.html#359" class="Generalizable">B</a><a id="594" class="Symbol">}</a> <a id="596" class="Symbol">→</a> <a id="598" href="Categories.Diagram.Pushout.html#571" class="Bound">h₁</a> <a id="601" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="603" href="Categories.Diagram.Pushout.html#412" class="Bound">f</a> <a id="605" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="607" href="Categories.Diagram.Pushout.html#584" class="Bound">h₂</a> <a id="610" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="612" href="Categories.Diagram.Pushout.html#424" class="Bound">g</a> <a id="614" class="Symbol">→</a> <a id="616" href="Categories.Diagram.Pushout.html#472" class="Field">Q</a> <a id="618" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="620" href="Categories.Diagram.Pushout.html#359" class="Generalizable">B</a>
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<a id="Pushout.unique"></a><a id="626" href="Categories.Diagram.Pushout.html#626" class="Field">unique</a> <a id="636" class="Symbol">:</a> <a id="638" class="Symbol">{</a><a id="639" href="Categories.Diagram.Pushout.html#639" class="Bound">h₁</a> <a id="642" class="Symbol">:</a> <a id="644" href="Categories.Diagram.Pushout.html#420" class="Bound">Y</a> <a id="646" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="648" href="Categories.Diagram.Pushout.html#361" class="Generalizable">E</a><a id="649" class="Symbol">}</a> <a id="651" class="Symbol">{</a><a id="652" href="Categories.Diagram.Pushout.html#652" class="Bound">h₂</a> <a id="655" class="Symbol">:</a> <a id="657" href="Categories.Diagram.Pushout.html#432" class="Bound">Z</a> <a id="659" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="661" href="Categories.Diagram.Pushout.html#361" class="Generalizable">E</a><a id="662" class="Symbol">}</a> <a id="664" class="Symbol">{</a><a id="665" href="Categories.Diagram.Pushout.html#665" class="Bound">j</a> <a id="667" class="Symbol">:</a> <a id="669" href="Categories.Diagram.Pushout.html#472" class="Field">Q</a> <a id="671" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="673" href="Categories.Diagram.Pushout.html#361" class="Generalizable">E</a><a id="674" class="Symbol">}</a> <a id="676" class="Symbol">{</a><a id="677" href="Categories.Diagram.Pushout.html#677" class="Bound">eq</a> <a id="680" class="Symbol">:</a> <a id="682" href="Categories.Diagram.Pushout.html#639" class="Bound">h₁</a> <a id="685" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="687" href="Categories.Diagram.Pushout.html#412" class="Bound">f</a> <a id="689" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="691" href="Categories.Diagram.Pushout.html#652" class="Bound">h₂</a> <a id="694" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="696" href="Categories.Diagram.Pushout.html#424" class="Bound">g</a><a id="697" class="Symbol">}</a> <a id="699" class="Symbol">→</a>
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<a id="719" href="Categories.Diagram.Pushout.html#665" class="Bound">j</a> <a id="721" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="723" href="Categories.Diagram.Pushout.html#485" class="Field">i₁</a> <a id="726" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="728" href="Categories.Diagram.Pushout.html#639" class="Bound">h₁</a> <a id="731" class="Symbol">→</a> <a id="733" href="Categories.Diagram.Pushout.html#665" class="Bound">j</a> <a id="735" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="737" href="Categories.Diagram.Pushout.html#501" class="Field">i₂</a> <a id="740" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="742" href="Categories.Diagram.Pushout.html#652" class="Bound">h₂</a> <a id="745" class="Symbol">→</a>
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<a id="765" href="Categories.Diagram.Pushout.html#665" class="Bound">j</a> <a id="767" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="769" href="Categories.Diagram.Pushout.html#558" class="Field">universal</a> <a id="779" href="Categories.Diagram.Pushout.html#677" class="Bound">eq</a>
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<a id="Pushout.universal∘i₁≈h₁"></a><a id="787" href="Categories.Diagram.Pushout.html#787" class="Field">universal∘i₁≈h₁</a> <a id="804" class="Symbol">:</a> <a id="806" class="Symbol">{</a><a id="807" href="Categories.Diagram.Pushout.html#807" class="Bound">h₁</a> <a id="810" class="Symbol">:</a> <a id="812" href="Categories.Diagram.Pushout.html#420" class="Bound">Y</a> <a id="814" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="816" href="Categories.Diagram.Pushout.html#361" class="Generalizable">E</a><a id="817" class="Symbol">}</a> <a id="819" class="Symbol">{</a><a id="820" href="Categories.Diagram.Pushout.html#820" class="Bound">h₂</a> <a id="823" class="Symbol">:</a> <a id="825" href="Categories.Diagram.Pushout.html#432" class="Bound">Z</a> <a id="827" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="829" href="Categories.Diagram.Pushout.html#361" class="Generalizable">E</a><a id="830" class="Symbol">}</a> <a id="832" class="Symbol">{</a><a id="833" href="Categories.Diagram.Pushout.html#833" class="Bound">eq</a> <a id="836" class="Symbol">:</a> <a id="838" href="Categories.Diagram.Pushout.html#807" class="Bound">h₁</a> <a id="841" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="843" href="Categories.Diagram.Pushout.html#412" class="Bound">f</a> <a id="845" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="847" href="Categories.Diagram.Pushout.html#820" class="Bound">h₂</a> <a id="850" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="852" href="Categories.Diagram.Pushout.html#424" class="Bound">g</a><a id="853" class="Symbol">}</a> <a id="855" class="Symbol">→</a>
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<a id="882" href="Categories.Diagram.Pushout.html#558" class="Field">universal</a> <a id="892" href="Categories.Diagram.Pushout.html#833" class="Bound">eq</a> <a id="895" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="897" href="Categories.Diagram.Pushout.html#485" class="Field">i₁</a> <a id="900" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="902" href="Categories.Diagram.Pushout.html#807" class="Bound">h₁</a>
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<a id="Pushout.universal∘i₂≈h₂"></a><a id="909" href="Categories.Diagram.Pushout.html#909" class="Field">universal∘i₂≈h₂</a> <a id="926" class="Symbol">:</a> <a id="928" class="Symbol">{</a><a id="929" href="Categories.Diagram.Pushout.html#929" class="Bound">h₁</a> <a id="932" class="Symbol">:</a> <a id="934" href="Categories.Diagram.Pushout.html#420" class="Bound">Y</a> <a id="936" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="938" href="Categories.Diagram.Pushout.html#361" class="Generalizable">E</a><a id="939" class="Symbol">}</a> <a id="941" class="Symbol">{</a><a id="942" href="Categories.Diagram.Pushout.html#942" class="Bound">h₂</a> <a id="945" class="Symbol">:</a> <a id="947" href="Categories.Diagram.Pushout.html#432" class="Bound">Z</a> <a id="949" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="951" href="Categories.Diagram.Pushout.html#361" class="Generalizable">E</a><a id="952" class="Symbol">}</a> <a id="954" class="Symbol">{</a><a id="955" href="Categories.Diagram.Pushout.html#955" class="Bound">eq</a> <a id="958" class="Symbol">:</a> <a id="960" href="Categories.Diagram.Pushout.html#929" class="Bound">h₁</a> <a id="963" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="965" href="Categories.Diagram.Pushout.html#412" class="Bound">f</a> <a id="967" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="969" href="Categories.Diagram.Pushout.html#942" class="Bound">h₂</a> <a id="972" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="974" href="Categories.Diagram.Pushout.html#424" class="Bound">g</a><a id="975" class="Symbol">}</a> <a id="977" class="Symbol">→</a>
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<a id="1004" href="Categories.Diagram.Pushout.html#558" class="Field">universal</a> <a id="1014" href="Categories.Diagram.Pushout.html#955" class="Bound">eq</a> <a id="1017" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1019" href="Categories.Diagram.Pushout.html#501" class="Field">i₂</a> <a id="1022" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1024" href="Categories.Diagram.Pushout.html#942" class="Bound">h₂</a>
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<a id="Pushout.unique-diagram"></a><a id="1030" href="Categories.Diagram.Pushout.html#1030" class="Function">unique-diagram</a> <a id="1045" class="Symbol">:</a> <a id="1047" href="Categories.Diagram.Pushout.html#383" class="Generalizable">h</a> <a id="1049" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1051" href="Categories.Diagram.Pushout.html#485" class="Field">i₁</a> <a id="1054" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1056" href="Categories.Diagram.Pushout.html#385" class="Generalizable">j</a> <a id="1058" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1060" href="Categories.Diagram.Pushout.html#485" class="Field">i₁</a> <a id="1063" class="Symbol">→</a>
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<a id="1084" href="Categories.Diagram.Pushout.html#383" class="Generalizable">h</a> <a id="1086" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1088" href="Categories.Diagram.Pushout.html#501" class="Field">i₂</a> <a id="1091" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1093" href="Categories.Diagram.Pushout.html#385" class="Generalizable">j</a> <a id="1095" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1097" href="Categories.Diagram.Pushout.html#501" class="Field">i₂</a> <a id="1100" class="Symbol">→</a>
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<a id="1121" href="Categories.Diagram.Pushout.html#383" class="Generalizable">h</a> <a id="1123" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="1125" href="Categories.Diagram.Pushout.html#385" class="Generalizable">j</a>
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<a id="1129" href="Categories.Diagram.Pushout.html#1030" class="Function">unique-diagram</a> <a id="1144" class="Symbol">{</a><a id="1145" class="Argument">h</a> <a id="1147" class="Symbol">=</a> <a id="1149" href="Categories.Diagram.Pushout.html#1149" class="Bound">h</a><a id="1150" class="Symbol">}</a> <a id="1152" class="Symbol">{</a><a id="1153" class="Argument">j</a> <a id="1155" class="Symbol">=</a> <a id="1157" href="Categories.Diagram.Pushout.html#1157" class="Bound">j</a><a id="1158" class="Symbol">}</a> <a id="1160" href="Categories.Diagram.Pushout.html#1160" class="Bound">eq₁</a> <a id="1164" href="Categories.Diagram.Pushout.html#1164" class="Bound">eq₂</a> <a id="1168" class="Symbol">=</a> <a id="1170" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
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<a id="1180" href="Categories.Diagram.Pushout.html#1149" class="Bound">h</a> <a id="1193" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1196" href="Categories.Diagram.Pushout.html#626" class="Field">unique</a> <a id="1203" href="Categories.Diagram.Pushout.html#1160" class="Bound">eq₁</a> <a id="1207" href="Categories.Diagram.Pushout.html#1164" class="Bound">eq₂</a> <a id="1211" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="1217" href="Categories.Diagram.Pushout.html#558" class="Field">universal</a> <a id="1227" href="Categories.Diagram.Pushout.html#1282" class="Function">eq</a> <a id="1230" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="1234" href="Categories.Diagram.Pushout.html#626" class="Field">unique</a> <a id="1241" href="Relation.Binary.Structures.html#1577" class="Function">refl</a> <a id="1246" href="Relation.Binary.Structures.html#1577" class="Function">refl</a> <a id="1251" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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<a id="1257" href="Categories.Diagram.Pushout.html#1157" class="Bound">j</a> <a id="1270" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
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<a id="1276" class="Keyword">where</a> <a id="1282" href="Categories.Diagram.Pushout.html#1282" class="Function">eq</a> <a id="1285" class="Symbol">=</a> <a id="1287" href="Categories.Morphism.Reasoning.Core.html#3392" class="Function">extendˡ</a> <a id="1295" href="Categories.Diagram.Pushout.html#526" class="Field">commute</a>
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</pre></body></html>
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