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<html><head><meta charset="utf-8"><title>Categories.Diagram.Pullback.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>
<a id="37" class="Keyword">open</a> <a id="42" class="Keyword">import</a> <a id="49" href="Categories.Category.html" class="Module">Categories.Category</a>
<a id="70" class="Keyword">module</a> <a id="77" href="Categories.Diagram.Pullback.Properties.html" class="Module">Categories.Diagram.Pullback.Properties</a> <a id="116" class="Symbol">{</a><a id="117" href="Categories.Diagram.Pullback.Properties.html#117" class="Bound">o</a> <a id="119" href="Categories.Diagram.Pullback.Properties.html#119" class="Bound">ℓ</a> <a id="121" href="Categories.Diagram.Pullback.Properties.html#121" class="Bound">e</a><a id="122" class="Symbol">}</a> <a id="124" class="Symbol">(</a><a id="125" href="Categories.Diagram.Pullback.Properties.html#125" class="Bound">C</a> <a id="127" class="Symbol">:</a> <a id="129" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="138" href="Categories.Diagram.Pullback.Properties.html#117" class="Bound">o</a> <a id="140" href="Categories.Diagram.Pullback.Properties.html#119" class="Bound">ℓ</a> <a id="142" href="Categories.Diagram.Pullback.Properties.html#121" class="Bound">e</a><a id="143" class="Symbol">)</a> <a id="145" class="Keyword">where</a>
<a id="152" class="Keyword">open</a> <a id="157" class="Keyword">import</a> <a id="164" href="Function.html" class="Module">Function</a> <a id="173" class="Keyword">using</a> <a id="179" class="Symbol">(</a><a id="180" href="Function.Base.html#1928" class="Function Operator">_$_</a><a id="183" class="Symbol">)</a>
<a id="186" class="Keyword">open</a> <a id="191" class="Keyword">import</a> <a id="198" href="Categories.Category.BinaryProducts.html" class="Module">Categories.Category.BinaryProducts</a> <a id="233" href="Categories.Diagram.Pullback.Properties.html#125" class="Bound">C</a>
<a id="235" class="Keyword">open</a> <a id="240" class="Keyword">import</a> <a id="247" href="Categories.Category.Cartesian.html" class="Module">Categories.Category.Cartesian</a> <a id="277" href="Categories.Diagram.Pullback.Properties.html#125" class="Bound">C</a>
<a id="279" class="Keyword">open</a> <a id="284" class="Keyword">import</a> <a id="291" href="Categories.Diagram.Pullback.html" class="Module">Categories.Diagram.Pullback</a> <a id="319" href="Categories.Diagram.Pullback.Properties.html#125" class="Bound">C</a>
<a id="321" class="Keyword">open</a> <a id="326" class="Keyword">import</a> <a id="333" href="Categories.Diagram.Equalizer.html" class="Module">Categories.Diagram.Equalizer</a> <a id="362" href="Categories.Diagram.Pullback.Properties.html#125" class="Bound">C</a> <a id="364" class="Keyword">hiding</a> <a id="371" class="Symbol">(</a><a id="372" href="Categories.Diagram.Equalizer.html#2254" class="Function">up-to-iso</a><a id="381" class="Symbol">)</a>
<a id="383" class="Keyword">open</a> <a id="388" class="Keyword">import</a> <a id="395" href="Categories.Object.Product.html" class="Module">Categories.Object.Product</a> <a id="421" href="Categories.Diagram.Pullback.Properties.html#125" class="Bound">C</a> <a id="423" class="Keyword">hiding</a> <a id="430" class="Symbol">(</a><a id="431" href="Categories.Object.Product.Core.html#2581" class="Function">up-to-iso</a><a id="440" class="Symbol">)</a>
<a id="442" class="Keyword">open</a> <a id="447" class="Keyword">import</a> <a id="454" href="Categories.Object.Terminal.html" class="Module">Categories.Object.Terminal</a> <a id="481" href="Categories.Diagram.Pullback.Properties.html#125" class="Bound">C</a> <a id="483" class="Keyword">hiding</a> <a id="490" class="Symbol">(</a><a id="491" href="Categories.Object.Terminal.html#1121" class="Function">up-to-iso</a><a id="500" class="Symbol">)</a>
<a id="502" class="Keyword">open</a> <a id="507" class="Keyword">import</a> <a id="514" href="Categories.Morphism.html" class="Module">Categories.Morphism</a> <a id="534" href="Categories.Diagram.Pullback.Properties.html#125" class="Bound">C</a>
<a id="536" class="Keyword">open</a> <a id="541" class="Keyword">import</a> <a id="548" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="578" href="Categories.Diagram.Pullback.Properties.html#125" class="Bound">C</a>
<a id="580" class="Keyword">open</a> <a id="585" class="Keyword">import</a> <a id="592" href="Categories.Category.Complete.Finitely.html" class="Module">Categories.Category.Complete.Finitely</a> <a id="630" class="Keyword">using</a> <a id="636" class="Symbol">(</a><a id="637" href="Categories.Category.Complete.Finitely.html#477" class="Record">FinitelyComplete</a><a id="653" class="Symbol">)</a>
<a id="655" class="Keyword">open</a> <a id="660" class="Keyword">import</a> <a id="667" href="Data.Product.html" class="Module">Data.Product</a> <a id="680" class="Keyword">using</a> <a id="686" class="Symbol">(</a><a id="687" href="Data.Product.html#1378" class="Function">∃</a><a id="688" class="Symbol">;</a> <a id="690" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">_,_</a><a id="693" class="Symbol">)</a>
<a id="696" class="Keyword">private</a>
<a id="706" class="Keyword">open</a> <a id="711" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="720" href="Categories.Diagram.Pullback.Properties.html#125" class="Bound">C</a>
<a id="724" class="Keyword">variable</a>
<a id="737" href="Categories.Diagram.Pullback.Properties.html#737" class="Generalizable">X</a> <a id="739" href="Categories.Diagram.Pullback.Properties.html#739" class="Generalizable">Y</a> <a id="741" href="Categories.Diagram.Pullback.Properties.html#741" class="Generalizable">Z</a> <a id="743" class="Symbol">:</a> <a id="745" href="Categories.Category.Core.html#559" class="Field">Obj</a>
<a id="753" href="Categories.Diagram.Pullback.Properties.html#753" class="Generalizable">f</a> <a id="755" href="Categories.Diagram.Pullback.Properties.html#755" class="Generalizable">g</a> <a id="757" href="Categories.Diagram.Pullback.Properties.html#757" class="Generalizable">h</a> <a id="759" href="Categories.Diagram.Pullback.Properties.html#759" class="Generalizable">i</a> <a id="761" class="Symbol">:</a> <a id="763" href="Categories.Diagram.Pullback.Properties.html#737" class="Generalizable">X</a> <a id="765" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="767" href="Categories.Diagram.Pullback.Properties.html#739" class="Generalizable">Y</a>
<a id="769" class="Keyword">open</a> <a id="774" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="787" class="Keyword">open</a> <a id="792" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="799" class="Comment">-- pullbacks of a monomorphism along itself give us the identity arrow.</a>
<a id="pullback-self-mono"></a><a id="871" href="Categories.Diagram.Pullback.Properties.html#871" class="Function">pullback-self-mono</a> <a id="890" class="Symbol">:</a> <a id="892" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="897" href="Categories.Diagram.Pullback.Properties.html#753" class="Generalizable">f</a> <a id="899" class="Symbol">→</a> <a id="901" href="Categories.Diagram.Pullback.html#747" class="Record">IsPullback</a> <a id="912" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="915" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="918" href="Categories.Diagram.Pullback.Properties.html#753" class="Generalizable">f</a> <a id="920" href="Categories.Diagram.Pullback.Properties.html#753" class="Generalizable">f</a>
<a id="922" href="Categories.Diagram.Pullback.Properties.html#871" class="Function">pullback-self-mono</a> <a id="941" href="Categories.Diagram.Pullback.Properties.html#941" class="Bound">mono</a> <a id="946" class="Symbol">=</a> <a id="948" class="Keyword">record</a>
<a id="957" class="Symbol">{</a> <a id="959" href="Categories.Diagram.Pullback.html#854" class="Field">commute</a> <a id="967" class="Symbol">=</a> <a id="969" href="Relation.Binary.Structures.html#1577" class="Function">refl</a>
<a id="976" class="Symbol">;</a> <a id="978" href="Categories.Diagram.Pullback.html#886" class="Field">universal</a> <a id="988" class="Symbol">=</a> <a id="990" class="Symbol">λ</a> <a id="992" class="Symbol">{</a><a id="993" href="Categories.Diagram.Pullback.Properties.html#993" class="Bound">X</a><a id="994" class="Symbol">}</a> <a id="996" class="Symbol">{</a><a id="997" href="Categories.Diagram.Pullback.Properties.html#997" class="Bound">h₁</a><a id="999" class="Symbol">}</a> <a id="1001" class="Symbol">{</a><a id="1002" href="Categories.Diagram.Pullback.Properties.html#1002" class="Bound">h₂</a><a id="1004" class="Symbol">}</a> <a id="1006" href="Categories.Diagram.Pullback.Properties.html#1006" class="Bound">eq</a> <a id="1009" class="Symbol">→</a> <a id="1011" href="Categories.Diagram.Pullback.Properties.html#997" class="Bound">h₁</a>
<a id="1016" class="Symbol">;</a> <a id="1018" href="Categories.Diagram.Pullback.html#956" class="Field">unique</a> <a id="1025" class="Symbol">=</a> <a id="1027" class="Symbol">λ</a> <a id="1029" href="Categories.Diagram.Pullback.Properties.html#1029" class="Bound">id∘i≈h₁</a> <a id="1037" href="Categories.Diagram.Pullback.Properties.html#1037" class="Bound">_</a> <a id="1039" class="Symbol">→</a> <a id="1041" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="1043" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="1053" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1055" href="Categories.Diagram.Pullback.Properties.html#1029" class="Bound">id∘i≈h₁</a>
<a id="1065" class="Symbol">;</a> <a id="1067" href="Categories.Diagram.Pullback.html#1081" class="Field">p₁∘universal≈h₁</a> <a id="1083" class="Symbol">=</a> <a id="1085" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a>
<a id="1097" class="Symbol">;</a> <a id="1099" href="Categories.Diagram.Pullback.html#1179" class="Field">p₂∘universal≈h₂</a> <a id="1115" class="Symbol">=</a> <a id="1117" class="Symbol">λ</a> <a id="1119" class="Symbol">{</a><a id="1120" href="Categories.Diagram.Pullback.Properties.html#1120" class="Bound">X</a><a id="1121" class="Symbol">}</a> <a id="1123" class="Symbol">{</a><a id="1124" href="Categories.Diagram.Pullback.Properties.html#1124" class="Bound">h₁</a><a id="1126" class="Symbol">}</a> <a id="1128" class="Symbol">{</a><a id="1129" href="Categories.Diagram.Pullback.Properties.html#1129" class="Bound">h₂</a><a id="1131" class="Symbol">}</a> <a id="1133" class="Symbol">{</a><a id="1134" href="Categories.Diagram.Pullback.Properties.html#1134" class="Bound">eq</a><a id="1136" class="Symbol">}</a> <a id="1138" class="Symbol">→</a> <a id="1140" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="1150" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1152" href="Categories.Diagram.Pullback.Properties.html#941" class="Bound">mono</a> <a id="1157" href="Categories.Diagram.Pullback.Properties.html#1124" class="Bound">h₁</a> <a id="1160" href="Categories.Diagram.Pullback.Properties.html#1129" class="Bound">h₂</a> <a id="1163" href="Categories.Diagram.Pullback.Properties.html#1134" class="Bound">eq</a>
<a id="1168" class="Symbol">}</a>
<a id="1171" class="Comment">-- pullback from a terminal object is the same as a product</a>
<a id="1231" class="Keyword">module</a> <a id="1238" href="Categories.Diagram.Pullback.Properties.html#1238" class="Module">_</a> <a id="1240" class="Symbol">(</a><a id="1241" href="Categories.Diagram.Pullback.Properties.html#1241" class="Bound">t</a> <a id="1243" class="Symbol">:</a> <a id="1245" href="Categories.Object.Terminal.html#860" class="Record">Terminal</a><a id="1253" class="Symbol">)</a> <a id="1255" class="Keyword">where</a>
<a id="1263" class="Keyword">open</a> <a id="1268" href="Categories.Object.Terminal.html#860" class="Module">Terminal</a> <a id="1277" href="Categories.Diagram.Pullback.Properties.html#1241" class="Bound">t</a>
<a id="1282" href="Categories.Diagram.Pullback.Properties.html#1282" class="Function">pullback-⊤⇒product</a> <a id="1301" class="Symbol">:</a> <a id="1303" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="1312" class="Symbol">(</a><a id="1313" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="1315" class="Symbol">{</a><a id="1316" href="Categories.Diagram.Pullback.Properties.html#737" class="Generalizable">X</a><a id="1317" class="Symbol">})</a> <a id="1320" class="Symbol">(</a><a id="1321" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="1323" class="Symbol">{</a><a id="1324" href="Categories.Diagram.Pullback.Properties.html#739" class="Generalizable">Y</a><a id="1325" class="Symbol">})</a> <a id="1328" class="Symbol">→</a> <a id="1330" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="1338" href="Categories.Diagram.Pullback.Properties.html#737" class="Generalizable">X</a> <a id="1340" href="Categories.Diagram.Pullback.Properties.html#739" class="Generalizable">Y</a>
<a id="1344" href="Categories.Diagram.Pullback.Properties.html#1282" class="Function">pullback-⊤⇒product</a> <a id="1363" href="Categories.Diagram.Pullback.Properties.html#1363" class="Bound">p</a> <a id="1365" class="Symbol">=</a> <a id="1367" class="Keyword">record</a>
<a id="1378" class="Symbol">{</a> <a id="1380" href="Categories.Object.Product.Core.html#510" class="Field">A×B</a> <a id="1389" class="Symbol">=</a> <a id="1391" href="Categories.Diagram.Pullback.html#2099" class="Field">P</a>
<a id="1397" class="Symbol">;</a> <a id="1399" href="Categories.Object.Product.Core.html#526" class="Field">π₁</a> <a id="1408" class="Symbol">=</a> <a id="1410" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a>
<a id="1417" class="Symbol">;</a> <a id="1419" href="Categories.Object.Product.Core.html#546" class="Field">π₂</a> <a id="1428" class="Symbol">=</a> <a id="1430" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a>
<a id="1437" class="Symbol">;</a> <a id="1439" href="Categories.Object.Product.Core.html#566" class="Field Operator">⟨_,_⟩</a> <a id="1448" class="Symbol">=</a> <a id="1450" class="Symbol">λ</a> <a id="1452" href="Categories.Diagram.Pullback.Properties.html#1452" class="Bound">f</a> <a id="1454" href="Categories.Diagram.Pullback.Properties.html#1454" class="Bound">g</a> <a id="1456" class="Symbol">→</a> <a id="1458" href="Categories.Diagram.Pullback.html#886" class="Function">universal</a> <a id="1468" class="Symbol">(</a><a id="1469" href="Categories.Object.Terminal.html#647" class="Function">!-unique₂</a> <a id="1479" class="Symbol">{</a><a id="1480" class="Argument">f</a> <a id="1482" class="Symbol">=</a> <a id="1484" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="1486" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1488" href="Categories.Diagram.Pullback.Properties.html#1452" class="Bound">f</a><a id="1489" class="Symbol">}</a> <a id="1491" class="Symbol">{</a><a id="1492" class="Argument">g</a> <a id="1494" class="Symbol">=</a> <a id="1496" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="1498" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1500" href="Categories.Diagram.Pullback.Properties.html#1454" class="Bound">g</a><a id="1501" class="Symbol">})</a>
<a id="1508" class="Symbol">;</a> <a id="1510" href="Categories.Object.Product.Core.html#603" class="Field">project₁</a> <a id="1519" class="Symbol">=</a> <a id="1521" href="Categories.Diagram.Pullback.html#1081" class="Function">p₁∘universal≈h₁</a>
<a id="1541" class="Symbol">;</a> <a id="1543" href="Categories.Object.Product.Core.html#637" class="Field">project₂</a> <a id="1552" class="Symbol">=</a> <a id="1554" href="Categories.Diagram.Pullback.html#1179" class="Function">p₂∘universal≈h₂</a>
<a id="1574" class="Symbol">;</a> <a id="1576" href="Categories.Object.Product.Core.html#671" class="Field">unique</a> <a id="1585" class="Symbol">=</a> <a id="1587" class="Symbol">λ</a> <a id="1589" href="Categories.Diagram.Pullback.Properties.html#1589" class="Bound">eq</a> <a id="1592" href="Categories.Diagram.Pullback.Properties.html#1592" class="Bound">eq′</a> <a id="1596" class="Symbol">→</a> <a id="1598" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="1600" class="Symbol">(</a><a id="1601" href="Categories.Diagram.Pullback.html#956" class="Function">unique</a> <a id="1608" href="Categories.Diagram.Pullback.Properties.html#1589" class="Bound">eq</a> <a id="1611" href="Categories.Diagram.Pullback.Properties.html#1592" class="Bound">eq′</a><a id="1614" class="Symbol">)</a>
<a id="1620" class="Symbol">}</a>
<a id="1626" class="Keyword">where</a> <a id="1632" class="Keyword">open</a> <a id="1637" href="Categories.Diagram.Pullback.html#2029" class="Module">Pullback</a> <a id="1646" href="Categories.Diagram.Pullback.Properties.html#1363" class="Bound">p</a>
<a id="1651" href="Categories.Diagram.Pullback.Properties.html#1651" class="Function">product⇒pullback-⊤</a> <a id="1670" class="Symbol">:</a> <a id="1672" href="Categories.Object.Product.Core.html#435" class="Record">Product</a> <a id="1680" href="Categories.Diagram.Pullback.Properties.html#737" class="Generalizable">X</a> <a id="1682" href="Categories.Diagram.Pullback.Properties.html#739" class="Generalizable">Y</a> <a id="1684" class="Symbol">→</a> <a id="1686" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="1695" class="Symbol">(</a><a id="1696" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="1698" class="Symbol">{</a><a id="1699" href="Categories.Diagram.Pullback.Properties.html#737" class="Generalizable">X</a><a id="1700" class="Symbol">})</a> <a id="1703" class="Symbol">(</a><a id="1704" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="1706" class="Symbol">{</a><a id="1707" href="Categories.Diagram.Pullback.Properties.html#739" class="Generalizable">Y</a><a id="1708" class="Symbol">})</a>
<a id="1713" href="Categories.Diagram.Pullback.Properties.html#1651" class="Function">product⇒pullback-⊤</a> <a id="1732" href="Categories.Diagram.Pullback.Properties.html#1732" class="Bound">p</a> <a id="1734" class="Symbol">=</a> <a id="1736" class="Keyword">record</a>
<a id="1747" class="Symbol">{</a> <a id="1749" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="1765" class="Symbol">=</a> <a id="1767" href="Categories.Object.Product.Core.html#526" class="Field">π₁</a>
<a id="1774" class="Symbol">;</a> <a id="1776" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="1792" class="Symbol">=</a> <a id="1794" href="Categories.Object.Product.Core.html#546" class="Field">π₂</a>
<a id="1801" class="Symbol">;</a> <a id="1803" href="Categories.Diagram.Pullback.html#2144" class="Field">isPullback</a> <a id="1814" class="Symbol">=</a> <a id="1816" class="Keyword">record</a>
<a id="1829" class="Symbol">{</a> <a id="1831" href="Categories.Diagram.Pullback.html#854" class="Field">commute</a> <a id="1847" class="Symbol">=</a> <a id="1849" href="Categories.Object.Terminal.html#647" class="Function">!-unique₂</a>
<a id="1865" class="Symbol">;</a> <a id="1867" href="Categories.Diagram.Pullback.html#886" class="Field">universal</a> <a id="1883" class="Symbol">=</a> <a id="1885" class="Symbol">λ</a> <a id="1887" class="Symbol">{</a><a id="1888" href="Categories.Diagram.Pullback.Properties.html#1888" class="Bound">_</a> <a id="1890" href="Categories.Diagram.Pullback.Properties.html#1890" class="Bound">f</a> <a id="1892" href="Categories.Diagram.Pullback.Properties.html#1892" class="Bound">g</a><a id="1893" class="Symbol">}</a> <a id="1895" href="Categories.Diagram.Pullback.Properties.html#1895" class="Bound">_</a> <a id="1897" class="Symbol">→</a> <a id="1899" href="Categories.Object.Product.Core.html#566" class="Field Operator">⟨</a> <a id="1901" href="Categories.Diagram.Pullback.Properties.html#1890" class="Bound">f</a> <a id="1903" href="Categories.Object.Product.Core.html#566" class="Field Operator">,</a> <a id="1905" href="Categories.Diagram.Pullback.Properties.html#1892" class="Bound">g</a> <a id="1907" href="Categories.Object.Product.Core.html#566" class="Field Operator">⟩</a>
<a id="1915" class="Symbol">;</a> <a id="1917" href="Categories.Diagram.Pullback.html#956" class="Field">unique</a> <a id="1933" class="Symbol">=</a> <a id="1935" class="Symbol">λ</a> <a id="1937" href="Categories.Diagram.Pullback.Properties.html#1937" class="Bound">eq</a> <a id="1940" href="Categories.Diagram.Pullback.Properties.html#1940" class="Bound">eq′</a> <a id="1944" class="Symbol">→</a> <a id="1946" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="1948" class="Symbol">(</a><a id="1949" href="Categories.Object.Product.Core.html#671" class="Field">unique</a> <a id="1956" href="Categories.Diagram.Pullback.Properties.html#1937" class="Bound">eq</a> <a id="1959" href="Categories.Diagram.Pullback.Properties.html#1940" class="Bound">eq′</a><a id="1962" class="Symbol">)</a>
<a id="1970" class="Symbol">;</a> <a id="1972" href="Categories.Diagram.Pullback.html#1081" class="Field">p₁∘universal≈h₁</a> <a id="1988" class="Symbol">=</a> <a id="1990" href="Categories.Object.Product.Core.html#603" class="Field">project₁</a>
<a id="2005" class="Symbol">;</a> <a id="2007" href="Categories.Diagram.Pullback.html#1179" class="Field">p₂∘universal≈h₂</a> <a id="2023" class="Symbol">=</a> <a id="2025" href="Categories.Object.Product.Core.html#637" class="Field">project₂</a>
<a id="2040" class="Symbol">}</a>
<a id="2046" class="Symbol">}</a>
<a id="2052" class="Keyword">where</a> <a id="2058" class="Keyword">open</a> <a id="2063" href="Categories.Object.Product.Core.html#435" class="Module">Product</a> <a id="2071" href="Categories.Diagram.Pullback.Properties.html#1732" class="Bound">p</a>
<a id="2074" class="Comment">-- pullbacks respect _≈_</a>
<a id="2099" class="Keyword">module</a> <a id="2106" href="Categories.Diagram.Pullback.Properties.html#2106" class="Module">_</a> <a id="2108" class="Symbol">(</a><a id="2109" href="Categories.Diagram.Pullback.Properties.html#2109" class="Bound">p</a> <a id="2111" class="Symbol">:</a> <a id="2113" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="2122" href="Categories.Diagram.Pullback.Properties.html#753" class="Generalizable">f</a> <a id="2124" href="Categories.Diagram.Pullback.Properties.html#755" class="Generalizable">g</a><a id="2125" class="Symbol">)</a> <a id="2127" class="Keyword">where</a>
<a id="2135" class="Keyword">open</a> <a id="2140" href="Categories.Diagram.Pullback.html#2029" class="Module">Pullback</a> <a id="2149" href="Categories.Diagram.Pullback.Properties.html#2109" class="Bound">p</a>
<a id="2154" href="Categories.Diagram.Pullback.Properties.html#2154" class="Function">pullback-resp-≈</a> <a id="2170" class="Symbol">:</a> <a id="2172" href="Categories.Diagram.Pullback.Properties.html#2122" class="Bound">f</a> <a id="2174" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="2176" href="Categories.Diagram.Pullback.Properties.html#757" class="Generalizable">h</a> <a id="2178" class="Symbol">→</a> <a id="2180" href="Categories.Diagram.Pullback.Properties.html#2124" class="Bound">g</a> <a id="2182" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="2184" href="Categories.Diagram.Pullback.Properties.html#759" class="Generalizable">i</a> <a id="2186" class="Symbol">→</a> <a id="2188" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="2197" href="Categories.Diagram.Pullback.Properties.html#757" class="Generalizable">h</a> <a id="2199" href="Categories.Diagram.Pullback.Properties.html#759" class="Generalizable">i</a>
<a id="2203" href="Categories.Diagram.Pullback.Properties.html#2154" class="Function">pullback-resp-≈</a> <a id="2219" href="Categories.Diagram.Pullback.Properties.html#2219" class="Bound">eq</a> <a id="2222" href="Categories.Diagram.Pullback.Properties.html#2222" class="Bound">eq′</a> <a id="2226" class="Symbol">=</a> <a id="2228" class="Keyword">record</a>
<a id="2239" class="Symbol">{</a> <a id="2241" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="2257" class="Symbol">=</a> <a id="2259" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a>
<a id="2266" class="Symbol">;</a> <a id="2268" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="2284" class="Symbol">=</a> <a id="2286" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a>
<a id="2293" class="Symbol">;</a> <a id="2295" href="Categories.Diagram.Pullback.html#2144" class="Field">isPullback</a> <a id="2306" class="Symbol">=</a> <a id="2308" class="Keyword">record</a>
<a id="2321" class="Symbol">{</a> <a id="2323" href="Categories.Diagram.Pullback.html#854" class="Field">commute</a> <a id="2339" class="Symbol">=</a> <a id="2341" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="2351" class="Symbol">(</a><a id="2352" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="2354" href="Categories.Diagram.Pullback.Properties.html#2219" class="Bound">eq</a><a id="2356" class="Symbol">)</a> <a id="2358" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2360" href="Categories.Diagram.Pullback.html#854" class="Function">commute</a> <a id="2368" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2370" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="2380" href="Categories.Diagram.Pullback.Properties.html#2222" class="Bound">eq′</a>
<a id="2390" class="Symbol">;</a> <a id="2392" href="Categories.Diagram.Pullback.html#886" class="Field">universal</a> <a id="2408" class="Symbol">=</a> <a id="2410" class="Symbol">λ</a> <a id="2412" href="Categories.Diagram.Pullback.Properties.html#2412" class="Bound">eq″</a> <a id="2416" class="Symbol">→</a> <a id="2418" href="Categories.Diagram.Pullback.html#886" class="Function">universal</a> <a id="2428" class="Symbol">(</a><a id="2429" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="2439" href="Categories.Diagram.Pullback.Properties.html#2219" class="Bound">eq</a> <a id="2442" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2444" href="Categories.Diagram.Pullback.Properties.html#2412" class="Bound">eq″</a> <a id="2448" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2450" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="2460" class="Symbol">(</a><a id="2461" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="2463" href="Categories.Diagram.Pullback.Properties.html#2222" class="Bound">eq′</a><a id="2466" class="Symbol">))</a>
<a id="2475" class="Symbol">;</a> <a id="2477" href="Categories.Diagram.Pullback.html#956" class="Field">unique</a> <a id="2493" class="Symbol">=</a> <a id="2495" href="Categories.Diagram.Pullback.html#956" class="Function">unique</a>
<a id="2508" class="Symbol">;</a> <a id="2510" href="Categories.Diagram.Pullback.html#1081" class="Field">p₁∘universal≈h₁</a> <a id="2526" class="Symbol">=</a> <a id="2528" href="Categories.Diagram.Pullback.html#1081" class="Function">p₁∘universal≈h₁</a>
<a id="2550" class="Symbol">;</a> <a id="2552" href="Categories.Diagram.Pullback.html#1179" class="Field">p₂∘universal≈h₂</a> <a id="2568" class="Symbol">=</a> <a id="2570" href="Categories.Diagram.Pullback.html#1179" class="Function">p₂∘universal≈h₂</a>
<a id="2592" class="Symbol">}</a>
<a id="2598" class="Symbol">}</a>
<a id="2601" class="Comment">-- Some facts about pulling back along identity</a>
<a id="2649" class="Keyword">module</a> <a id="2656" href="Categories.Diagram.Pullback.Properties.html#2656" class="Module">_</a> <a id="2658" class="Symbol">(</a><a id="2659" href="Categories.Diagram.Pullback.Properties.html#2659" class="Bound">p</a> <a id="2661" class="Symbol">:</a> <a id="2663" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="2672" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="2675" href="Categories.Diagram.Pullback.Properties.html#753" class="Generalizable">f</a><a id="2676" class="Symbol">)</a> <a id="2678" class="Keyword">where</a>
<a id="2686" class="Keyword">open</a> <a id="2691" href="Categories.Diagram.Pullback.html#2029" class="Module">Pullback</a> <a id="2700" href="Categories.Diagram.Pullback.Properties.html#2659" class="Bound">p</a>
<a id="2705" class="Comment">-- This is a more subtle way of saying that &#39;p₂ ≈ id&#39;, without involving heterogenous equality.</a>
<a id="2803" href="Categories.Diagram.Pullback.Properties.html#2803" class="Function">pullback-identity</a> <a id="2821" class="Symbol">:</a> <a id="2823" href="Categories.Diagram.Pullback.html#886" class="Function">universal</a> <a id="2833" href="Categories.Morphism.Reasoning.Core.html#1309" class="Function">id-comm-sym</a> <a id="2845" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2847" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="2850" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="2852" href="Categories.Category.Core.html#630" class="Field">id</a>
<a id="2857" href="Categories.Diagram.Pullback.Properties.html#2803" class="Function">pullback-identity</a> <a id="2875" class="Symbol">=</a> <a id="2877" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="2887" href="Categories.Diagram.Pullback.html#886" class="Function">universal</a> <a id="2897" href="Categories.Morphism.Reasoning.Core.html#1309" class="Function">id-comm-sym</a> <a id="2909" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2911" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="2914" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2917" href="Categories.Diagram.Pullback.html#956" class="Function">unique</a> <a id="2924" class="Symbol">(</a> <a id="2926" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="2932" href="Categories.Diagram.Pullback.html#1081" class="Function">p₁∘universal≈h₁</a> <a id="2948" class="Symbol">)</a> <a id="2950" class="Symbol">(</a><a id="2951" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="2957" href="Categories.Diagram.Pullback.html#1179" class="Function">p₂∘universal≈h₂</a><a id="2972" class="Symbol">)</a> <a id="2975" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2981" href="Categories.Diagram.Pullback.html#886" class="Function">universal</a> <a id="2991" href="Categories.Diagram.Pullback.Properties.html#3147" class="Function">eq</a> <a id="3008" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="3011" href="Categories.Diagram.Pullback.html#1484" class="Function">universal-resp-≈</a> <a id="3028" class="Symbol">(</a><a id="3029" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="3031" href="Categories.Diagram.Pullback.html#854" class="Function">commute</a> <a id="3039" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="3041" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a><a id="3050" class="Symbol">)</a> <a id="3052" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="3062" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="3068" href="Categories.Diagram.Pullback.html#886" class="Function">universal</a> <a id="3078" href="Categories.Diagram.Pullback.html#854" class="Function">commute</a> <a id="3095" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="3099" href="Categories.Diagram.Pullback.html#1405" class="Function">Pullback.id-unique</a> <a id="3118" href="Categories.Diagram.Pullback.Properties.html#2659" class="Bound">p</a> <a id="3120" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="3126" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="3129" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="3135" class="Keyword">where</a>
<a id="3147" href="Categories.Diagram.Pullback.Properties.html#3147" class="Function">eq</a> <a id="3150" class="Symbol">:</a> <a id="3152" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="3155" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3157" href="Categories.Diagram.Pullback.Properties.html#2675" class="Bound">f</a> <a id="3159" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3161" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="3164" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="3166" href="Categories.Diagram.Pullback.Properties.html#2675" class="Bound">f</a> <a id="3168" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3170" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="3173" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3175" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a>
<a id="3184" href="Categories.Diagram.Pullback.Properties.html#3147" class="Function">eq</a> <a id="3187" class="Symbol">=</a> <a id="3189" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="3203" class="Symbol">(</a><a id="3204" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="3207" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3209" href="Categories.Diagram.Pullback.Properties.html#2675" class="Bound">f</a> <a id="3211" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3213" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a><a id="3215" class="Symbol">)</a> <a id="3217" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="3220" href="Categories.Morphism.Reasoning.Core.html#2948" class="Function">elimˡ</a> <a id="3226" href="Relation.Binary.Structures.html#1577" class="Function">Equiv.refl</a> <a id="3237" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="3247" class="Symbol">(</a><a id="3248" href="Categories.Diagram.Pullback.Properties.html#2675" class="Bound">f</a> <a id="3250" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3252" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a><a id="3254" class="Symbol">)</a> <a id="3261" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="3265" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="3273" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="3283" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="3293" class="Symbol">(</a><a id="3294" href="Categories.Diagram.Pullback.Properties.html#2675" class="Bound">f</a> <a id="3296" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3298" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="3301" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3303" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a><a id="3305" class="Symbol">)</a> <a id="3307" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="3310" class="Comment">-- pullbacks in Cartesian categories create equalizers</a>
<a id="3365" class="Keyword">module</a> <a id="3372" href="Categories.Diagram.Pullback.Properties.html#3372" class="Module">_</a> <a id="3374" class="Symbol">(</a><a id="3375" href="Categories.Diagram.Pullback.Properties.html#3375" class="Bound">pullbacks</a> <a id="3385" class="Symbol">:</a> <a id="3387" class="Symbol">∀</a> <a id="3389" class="Symbol">{</a><a id="3390" href="Categories.Diagram.Pullback.Properties.html#3390" class="Bound">X</a> <a id="3392" href="Categories.Diagram.Pullback.Properties.html#3392" class="Bound">Y</a> <a id="3394" href="Categories.Diagram.Pullback.Properties.html#3394" class="Bound">Z</a><a id="3395" class="Symbol">}</a> <a id="3397" class="Symbol">(</a><a id="3398" href="Categories.Diagram.Pullback.Properties.html#3398" class="Bound">f</a> <a id="3400" class="Symbol">:</a> <a id="3402" href="Categories.Diagram.Pullback.Properties.html#3390" class="Bound">X</a> <a id="3404" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="3406" href="Categories.Diagram.Pullback.Properties.html#3394" class="Bound">Z</a><a id="3407" class="Symbol">)</a> <a id="3409" class="Symbol">(</a><a id="3410" href="Categories.Diagram.Pullback.Properties.html#3410" class="Bound">g</a> <a id="3412" class="Symbol">:</a> <a id="3414" href="Categories.Diagram.Pullback.Properties.html#3392" class="Bound">Y</a> <a id="3416" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="3418" href="Categories.Diagram.Pullback.Properties.html#3394" class="Bound">Z</a><a id="3419" class="Symbol">)</a> <a id="3421" class="Symbol">→</a> <a id="3423" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="3432" href="Categories.Diagram.Pullback.Properties.html#3398" class="Bound">f</a> <a id="3434" href="Categories.Diagram.Pullback.Properties.html#3410" class="Bound">g</a><a id="3435" class="Symbol">)</a>
<a id="3446" class="Symbol">(</a><a id="3447" href="Categories.Diagram.Pullback.Properties.html#3447" class="Bound">cartesian</a> <a id="3457" class="Symbol">:</a> <a id="3459" href="Categories.Category.Cartesian.html#727" class="Record">Cartesian</a><a id="3468" class="Symbol">)</a> <a id="3470" class="Keyword">where</a>
<a id="3478" class="Keyword">open</a> <a id="3483" href="Categories.Category.Cartesian.html#727" class="Module">Cartesian</a> <a id="3493" href="Categories.Diagram.Pullback.Properties.html#3447" class="Bound">cartesian</a>
<a id="3505" class="Keyword">open</a> <a id="3510" href="Categories.Category.BinaryProducts.html#812" class="Module">BinaryProducts</a> <a id="3525" href="Categories.Category.Cartesian.html#801" class="Field">products</a> <a id="3534" class="Keyword">using</a> <a id="3540" class="Symbol">(</a><a id="3541" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨_,_⟩</a><a id="3546" class="Symbol">;</a> <a id="3548" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a><a id="3550" class="Symbol">;</a> <a id="3552" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a><a id="3554" class="Symbol">;</a> <a id="3556" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a><a id="3564" class="Symbol">;</a> <a id="3566" href="Categories.Category.BinaryProducts.html#2939" class="Function">⟨⟩∘</a><a id="3569" class="Symbol">;</a> <a id="3571" href="Categories.Object.Product.Core.html#603" class="Function">project₁</a><a id="3579" class="Symbol">;</a> <a id="3581" href="Categories.Object.Product.Core.html#637" class="Function">project₂</a><a id="3589" class="Symbol">)</a>
<a id="3594" href="Categories.Diagram.Pullback.Properties.html#3594" class="Function">pullback×cartesian⇒equalizer</a> <a id="3623" class="Symbol">:</a> <a id="3625" href="Categories.Diagram.Equalizer.html#1948" class="Record">Equalizer</a> <a id="3635" href="Categories.Diagram.Pullback.Properties.html#753" class="Generalizable">f</a> <a id="3637" href="Categories.Diagram.Pullback.Properties.html#755" class="Generalizable">g</a>
<a id="3641" href="Categories.Diagram.Pullback.Properties.html#3594" class="Function">pullback×cartesian⇒equalizer</a> <a id="3670" class="Symbol">{</a><a id="3671" class="Argument">f</a> <a id="3673" class="Symbol">=</a> <a id="3675" href="Categories.Diagram.Pullback.Properties.html#3675" class="Bound">f</a><a id="3676" class="Symbol">}</a> <a id="3678" class="Symbol">{</a><a id="3679" class="Argument">g</a> <a id="3681" class="Symbol">=</a> <a id="3683" href="Categories.Diagram.Pullback.Properties.html#3683" class="Bound">g</a><a id="3684" class="Symbol">}</a> <a id="3686" class="Symbol">=</a> <a id="3688" class="Keyword">record</a>
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<a id="4448" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4450" href="Categories.Diagram.Pullback.Properties.html#3675" class="Bound">f</a> <a id="4452" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4454" href="Categories.Diagram.Pullback.Properties.html#3683" class="Bound">g</a> <a id="4456" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="4458" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4460" href="Categories.Diagram.Pullback.html#2112" class="Function">p.p₁</a> <a id="4477" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="4480" href="Categories.Diagram.Pullback.html#854" class="Function">p.commute</a> <a id="4490" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="4504" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4506" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4509" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4511" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4514" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="4516" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4518" href="Categories.Diagram.Pullback.html#2128" class="Function">p.p₂</a> <a id="4533" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="4536" href="Categories.Category.BinaryProducts.html#2939" class="Function">⟨⟩∘</a> <a id="4540" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="4554" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4556" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4559" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4561" href="Categories.Diagram.Pullback.html#2128" class="Function">p.p₂</a> <a id="4566" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4568" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4571" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4573" href="Categories.Diagram.Pullback.html#2128" class="Function">p.p₂</a> <a id="4578" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="4583" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="4586" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a> <a id="4595" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="4605" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="4615" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="4629" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4631" href="Categories.Diagram.Pullback.html#2128" class="Function">p.p₂</a> <a id="4636" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4638" href="Categories.Diagram.Pullback.html#2128" class="Function">p.p₂</a> <a id="4643" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="4658" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="4670" href="Categories.Diagram.Pullback.Properties.html#4670" class="Function">eq′</a> <a id="4674" class="Symbol">:</a> <a id="4676" href="Categories.Diagram.Pullback.Properties.html#3675" class="Bound">f</a> <a id="4678" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4680" href="Categories.Diagram.Pullback.html#2112" class="Function">p.p₁</a> <a id="4685" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="4688" href="Categories.Diagram.Pullback.html#2128" class="Function">p.p₂</a>
<a id="4703" href="Categories.Diagram.Pullback.Properties.html#4670" class="Function">eq′</a> <a id="4707" class="Symbol">=</a> <a id="4709" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="4727" href="Categories.Diagram.Pullback.Properties.html#3675" class="Bound">f</a> <a id="4729" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4731" href="Categories.Diagram.Pullback.html#2112" class="Function">p.p₁</a> <a id="4756" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="4760" href="Categories.Object.Product.Core.html#603" class="Function">project₁</a> <a id="4769" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="4783" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="4786" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4788" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4790" href="Categories.Diagram.Pullback.Properties.html#3675" class="Bound">f</a> <a id="4792" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4794" href="Categories.Diagram.Pullback.html#2112" class="Function">p.p₁</a> <a id="4799" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4801" href="Categories.Diagram.Pullback.Properties.html#3683" class="Bound">g</a> <a id="4803" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4805" href="Categories.Diagram.Pullback.html#2112" class="Function">p.p₁</a> <a id="4810" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="4812" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="4815" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="4823" href="Categories.Diagram.Pullback.Properties.html#4317" class="Function">eq</a> <a id="4826" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="4936" href="Categories.Diagram.Pullback.Properties.html#4936" class="Function">equality</a> <a id="4945" class="Symbol">:</a> <a id="4947" href="Categories.Diagram.Pullback.Properties.html#3675" class="Bound">f</a> <a id="4949" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4951" href="Categories.Diagram.Pullback.html#2112" class="Function">p.p₁</a> <a id="4956" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="4958" href="Categories.Diagram.Pullback.Properties.html#3683" class="Bound">g</a> <a id="4960" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4962" href="Categories.Diagram.Pullback.html#2112" class="Function">p.p₁</a>
<a id="4977" href="Categories.Diagram.Pullback.Properties.html#4936" class="Function">equality</a> <a id="4986" class="Symbol">=</a> <a id="4988" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="5006" href="Categories.Diagram.Pullback.Properties.html#3675" class="Bound">f</a> <a id="5008" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5010" href="Categories.Diagram.Pullback.html#2112" class="Function">p.p₁</a> <a id="5035" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="5038" href="Categories.Diagram.Pullback.Properties.html#4670" class="Function">eq′</a> <a id="5042" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="5056" href="Categories.Diagram.Pullback.html#2128" class="Function">p.p₂</a> <a id="5085" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5089" href="Categories.Object.Product.Core.html#637" class="Function">project₂</a> <a id="5098" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="5112" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="5115" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5117" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="5119" href="Categories.Diagram.Pullback.html#2128" class="Function">p.p₂</a> <a id="5124" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="5126" href="Categories.Diagram.Pullback.html#2128" class="Function">p.p₂</a> <a id="5131" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="5141" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5145" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="5153" href="Categories.Diagram.Pullback.Properties.html#4317" class="Function">eq</a> <a id="5156" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="5170" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="5173" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5175" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="5177" href="Categories.Diagram.Pullback.Properties.html#3675" class="Bound">f</a> <a id="5179" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5181" href="Categories.Diagram.Pullback.html#2112" class="Function">p.p₁</a> <a id="5186" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="5188" href="Categories.Diagram.Pullback.Properties.html#3683" class="Bound">g</a> <a id="5190" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5192" href="Categories.Diagram.Pullback.html#2112" class="Function">p.p₁</a> <a id="5197" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="5199" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="5202" href="Categories.Object.Product.Core.html#637" class="Function">project₂</a> <a id="5211" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="5225" href="Categories.Diagram.Pullback.Properties.html#3683" class="Bound">g</a> <a id="5227" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5229" href="Categories.Diagram.Pullback.html#2112" class="Function">p.p₁</a> <a id="5254" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="5257" class="Comment">-- all pullbacks and a terminal object make a category finitely complete</a>
<a id="pullback-⊤⇒FinitelyComplete"></a><a id="5330" href="Categories.Diagram.Pullback.Properties.html#5330" class="Function">pullback-⊤⇒FinitelyComplete</a> <a id="5358" class="Symbol">:</a> <a id="5360" class="Symbol">(∀</a> <a id="5363" class="Symbol">{</a><a id="5364" href="Categories.Diagram.Pullback.Properties.html#5364" class="Bound">X</a> <a id="5366" href="Categories.Diagram.Pullback.Properties.html#5366" class="Bound">Y</a> <a id="5368" href="Categories.Diagram.Pullback.Properties.html#5368" class="Bound">Z</a><a id="5369" class="Symbol">}</a> <a id="5371" class="Symbol">(</a><a id="5372" href="Categories.Diagram.Pullback.Properties.html#5372" class="Bound">f</a> <a id="5374" class="Symbol">:</a> <a id="5376" href="Categories.Diagram.Pullback.Properties.html#5364" class="Bound">X</a> <a id="5378" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="5380" href="Categories.Diagram.Pullback.Properties.html#5368" class="Bound">Z</a><a id="5381" class="Symbol">)</a> <a id="5383" class="Symbol">(</a><a id="5384" href="Categories.Diagram.Pullback.Properties.html#5384" class="Bound">g</a> <a id="5386" class="Symbol">:</a> <a id="5388" href="Categories.Diagram.Pullback.Properties.html#5366" class="Bound">Y</a> <a id="5390" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="5392" href="Categories.Diagram.Pullback.Properties.html#5368" class="Bound">Z</a><a id="5393" class="Symbol">)</a> <a id="5395" class="Symbol">→</a> <a id="5397" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="5406" href="Categories.Diagram.Pullback.Properties.html#5372" class="Bound">f</a> <a id="5408" href="Categories.Diagram.Pullback.Properties.html#5384" class="Bound">g</a><a id="5409" class="Symbol">)</a> <a id="5411" class="Symbol">→</a> <a id="5413" href="Categories.Object.Terminal.html#860" class="Record">Terminal</a> <a id="5422" class="Symbol">→</a> <a id="5424" href="Categories.Category.Complete.Finitely.html#477" class="Record">FinitelyComplete</a> <a id="5441" href="Categories.Diagram.Pullback.Properties.html#125" class="Bound">C</a>
<a id="5443" href="Categories.Diagram.Pullback.Properties.html#5330" class="Function">pullback-⊤⇒FinitelyComplete</a> <a id="5471" href="Categories.Diagram.Pullback.Properties.html#5471" class="Bound">pullbacks</a> <a id="5481" href="Categories.Diagram.Pullback.Properties.html#5481" class="Bound">⊤</a> <a id="5483" class="Symbol">=</a> <a id="5485" class="Keyword">record</a>
<a id="5494" class="Symbol">{</a> <a id="5496" href="Categories.Category.Complete.Finitely.html#534" class="Field">cartesian</a> <a id="5506" class="Symbol">=</a> <a id="5508" href="Categories.Diagram.Pullback.Properties.html#5874" class="Function">cartesian</a>
<a id="5520" class="Symbol">;</a> <a id="5522" href="Categories.Category.Complete.Finitely.html#560" class="Field">equalizer</a> <a id="5532" class="Symbol">=</a> <a id="5534" class="Symbol">λ</a> <a id="5536" href="Categories.Diagram.Pullback.Properties.html#5536" class="Bound">_</a> <a id="5538" href="Categories.Diagram.Pullback.Properties.html#5538" class="Bound">_</a> <a id="5540" class="Symbol">→</a> <a id="5542" href="Categories.Diagram.Pullback.Properties.html#3594" class="Function">pullback×cartesian⇒equalizer</a> <a id="5571" href="Categories.Diagram.Pullback.Properties.html#5471" class="Bound">pullbacks</a> <a id="5581" href="Categories.Diagram.Pullback.Properties.html#5874" class="Function">cartesian</a>
<a id="5593" class="Symbol">}</a>
<a id="5599" class="Keyword">where</a>
<a id="5611" class="Keyword">open</a> <a id="5616" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="5625" class="Keyword">hiding</a> <a id="5632" class="Symbol">(</a><a id="5633" href="Categories.Category.Core.html#559" class="Field">Obj</a><a id="5636" class="Symbol">)</a>
<a id="5644" class="Keyword">open</a> <a id="5649" href="Categories.Diagram.Pullback.html#2029" class="Module">Pullback</a>
<a id="5664" class="Keyword">open</a> <a id="5669" href="Categories.Object.Terminal.html#860" class="Module">Terminal</a> <a id="5678" href="Categories.Diagram.Pullback.Properties.html#5481" class="Bound">⊤</a> <a id="5680" class="Keyword">hiding</a> <a id="5687" class="Symbol">(</a><a id="5688" href="Categories.Object.Terminal.html#905" class="Field">⊤</a><a id="5689" class="Symbol">)</a>
<a id="5698" href="Categories.Diagram.Pullback.Properties.html#5698" class="Function Operator">_×_</a> <a id="5702" class="Symbol">:</a> <a id="5704" class="Symbol">(</a><a id="5705" href="Categories.Diagram.Pullback.Properties.html#5705" class="Bound">A</a> <a id="5707" href="Categories.Diagram.Pullback.Properties.html#5707" class="Bound">B</a> <a id="5709" class="Symbol">:</a> <a id="5711" href="Categories.Category.Core.html#559" class="Field">Obj</a><a id="5714" class="Symbol">)</a> <a id="5716" class="Symbol">→</a> <a id="5718" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="5727" class="Symbol">(</a><a id="5728" href="Categories.Object.Terminal.html#577" class="Field">IsTerminal.!</a> <a id="5741" href="Categories.Object.Terminal.html#917" class="Field">⊤-is-terminal</a><a id="5754" class="Symbol">)</a> <a id="5756" class="Symbol">(</a><a id="5757" href="Categories.Object.Terminal.html#577" class="Field">IsTerminal.!</a> <a id="5770" href="Categories.Object.Terminal.html#917" class="Field">⊤-is-terminal</a><a id="5783" class="Symbol">)</a>
<a id="5791" href="Categories.Diagram.Pullback.Properties.html#5791" class="Bound">A</a> <a id="5793" href="Categories.Diagram.Pullback.Properties.html#5698" class="Function Operator">×</a> <a id="5795" href="Categories.Diagram.Pullback.Properties.html#5795" class="Bound">B</a> <a id="5797" class="Symbol">=</a> <a id="5799" href="Categories.Diagram.Pullback.Properties.html#5471" class="Bound">pullbacks</a> <a id="5809" class="Symbol">(</a><a id="5810" href="Categories.Object.Terminal.html#577" class="Field">IsTerminal.!</a> <a id="5823" href="Categories.Object.Terminal.html#917" class="Field">⊤-is-terminal</a><a id="5836" class="Symbol">)</a> <a id="5838" class="Symbol">(</a><a id="5839" href="Categories.Object.Terminal.html#577" class="Field">IsTerminal.!</a> <a id="5852" href="Categories.Object.Terminal.html#917" class="Field">⊤-is-terminal</a><a id="5865" class="Symbol">)</a>
<a id="5874" href="Categories.Diagram.Pullback.Properties.html#5874" class="Function">cartesian</a> <a id="5884" class="Symbol">=</a> <a id="5886" class="Keyword">record</a>
<a id="5901" class="Symbol">{</a> <a id="5903" href="Categories.Category.Cartesian.html#777" class="Field">terminal</a> <a id="5912" class="Symbol">=</a> <a id="5914" href="Categories.Diagram.Pullback.Properties.html#5481" class="Bound">⊤</a>
<a id="5924" class="Symbol">;</a> <a id="5926" href="Categories.Category.Cartesian.html#801" class="Field">products</a> <a id="5935" class="Symbol">=</a> <a id="5937" class="Keyword">record</a>
<a id="5956" class="Symbol">{</a> <a id="5958" href="Categories.Category.BinaryProducts.html#916" class="Field">product</a> <a id="5966" class="Symbol">=</a> <a id="5968" class="Symbol">λ</a> <a id="5970" class="Symbol">{</a><a id="5971" href="Categories.Diagram.Pullback.Properties.html#5971" class="Bound">A</a> <a id="5973" href="Categories.Diagram.Pullback.Properties.html#5973" class="Bound">B</a><a id="5974" class="Symbol">}</a> <a id="5976" class="Symbol">→</a> <a id="5978" class="Keyword">record</a>
<a id="6001" class="Symbol">{</a> <a id="6003" href="Categories.Object.Product.Core.html#510" class="Field">A×B</a> <a id="6007" class="Symbol">=</a> <a id="6009" href="Categories.Diagram.Pullback.html#2099" class="Field">P</a> <a id="6011" class="Symbol">{</a><a id="6012" href="Categories.Diagram.Pullback.Properties.html#5971" class="Bound">A</a><a id="6013" class="Symbol">}{_}{</a><a id="6018" href="Categories.Diagram.Pullback.Properties.html#5973" class="Bound">B</a><a id="6019" class="Symbol">}</a> <a id="6021" class="Symbol">(</a><a id="6022" href="Categories.Diagram.Pullback.Properties.html#5971" class="Bound">A</a> <a id="6024" href="Categories.Diagram.Pullback.Properties.html#5698" class="Function Operator">×</a> <a id="6026" href="Categories.Diagram.Pullback.Properties.html#5973" class="Bound">B</a><a id="6027" class="Symbol">)</a>
<a id="6045" class="Symbol">;</a> <a id="6047" href="Categories.Object.Product.Core.html#526" class="Field">π₁</a> <a id="6050" class="Symbol">=</a> <a id="6052" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="6055" class="Symbol">(</a><a id="6056" href="Categories.Diagram.Pullback.Properties.html#5971" class="Bound">A</a> <a id="6058" href="Categories.Diagram.Pullback.Properties.html#5698" class="Function Operator">×</a> <a id="6060" href="Categories.Diagram.Pullback.Properties.html#5973" class="Bound">B</a><a id="6061" class="Symbol">)</a>
<a id="6079" class="Symbol">;</a> <a id="6081" href="Categories.Object.Product.Core.html#546" class="Field">π₂</a> <a id="6084" class="Symbol">=</a> <a id="6086" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="6089" class="Symbol">(</a><a id="6090" href="Categories.Diagram.Pullback.Properties.html#5971" class="Bound">A</a> <a id="6092" href="Categories.Diagram.Pullback.Properties.html#5698" class="Function Operator">×</a> <a id="6094" href="Categories.Diagram.Pullback.Properties.html#5973" class="Bound">B</a><a id="6095" class="Symbol">)</a>
<a id="6113" class="Symbol">;</a> <a id="6115" href="Categories.Object.Product.Core.html#566" class="Field Operator">⟨_,_⟩</a> <a id="6121" class="Symbol">=</a> <a id="6124" class="Symbol">λ</a> <a id="6126" href="Categories.Diagram.Pullback.Properties.html#6126" class="Bound">_</a> <a id="6128" href="Categories.Diagram.Pullback.Properties.html#6128" class="Bound">_</a> <a id="6130" class="Symbol">→</a> <a id="6132" href="Categories.Diagram.Pullback.html#886" class="Function">universal</a> <a id="6142" class="Symbol">(</a><a id="6143" href="Categories.Diagram.Pullback.Properties.html#5971" class="Bound">A</a> <a id="6145" href="Categories.Diagram.Pullback.Properties.html#5698" class="Function Operator">×</a> <a id="6147" href="Categories.Diagram.Pullback.Properties.html#5973" class="Bound">B</a><a id="6148" class="Symbol">)</a> <a id="6150" class="Symbol">(</a><a id="6151" href="Categories.Object.Terminal.html#647" class="Function">!-unique₂</a><a id="6160" class="Symbol">)</a>
<a id="6178" class="Symbol">;</a> <a id="6180" href="Categories.Object.Product.Core.html#603" class="Field">project₁</a> <a id="6189" class="Symbol">=</a> <a id="6191" href="Categories.Diagram.Pullback.html#1081" class="Function">p₁∘universal≈h₁</a> <a id="6207" class="Symbol">(</a><a id="6208" href="Categories.Diagram.Pullback.Properties.html#5971" class="Bound">A</a> <a id="6210" href="Categories.Diagram.Pullback.Properties.html#5698" class="Function Operator">×</a> <a id="6212" href="Categories.Diagram.Pullback.Properties.html#5973" class="Bound">B</a><a id="6213" class="Symbol">)</a>
<a id="6231" class="Symbol">;</a> <a id="6233" href="Categories.Object.Product.Core.html#637" class="Field">project₂</a> <a id="6242" class="Symbol">=</a> <a id="6244" href="Categories.Diagram.Pullback.html#1179" class="Function">p₂∘universal≈h₂</a> <a id="6260" class="Symbol">(</a><a id="6261" href="Categories.Diagram.Pullback.Properties.html#5971" class="Bound">A</a> <a id="6263" href="Categories.Diagram.Pullback.Properties.html#5698" class="Function Operator">×</a> <a id="6265" href="Categories.Diagram.Pullback.Properties.html#5973" class="Bound">B</a><a id="6266" class="Symbol">)</a>
<a id="6284" class="Symbol">;</a> <a id="6286" href="Categories.Object.Product.Core.html#671" class="Field">unique</a> <a id="6293" class="Symbol">=</a> <a id="6295" class="Symbol">λ</a> <a id="6297" href="Categories.Diagram.Pullback.Properties.html#6297" class="Bound">eq₁</a> <a id="6301" href="Categories.Diagram.Pullback.Properties.html#6301" class="Bound">eq₂</a> <a id="6305" class="Symbol">→</a> <a id="6307" href="Relation.Binary.Structures.html#1603" class="Field">Equiv.sym</a> <a id="6317" href="Categories.Diagram.Pullback.Properties.html#125" class="Bound">C</a> <a id="6319" class="Symbol">(</a><a id="6320" href="Categories.Diagram.Pullback.html#956" class="Function">unique</a> <a id="6327" class="Symbol">(</a><a id="6328" href="Categories.Diagram.Pullback.Properties.html#5971" class="Bound">A</a> <a id="6330" href="Categories.Diagram.Pullback.Properties.html#5698" class="Function Operator">×</a> <a id="6332" href="Categories.Diagram.Pullback.Properties.html#5973" class="Bound">B</a><a id="6333" class="Symbol">)</a> <a id="6335" href="Categories.Diagram.Pullback.Properties.html#6297" class="Bound">eq₁</a> <a id="6339" href="Categories.Diagram.Pullback.Properties.html#6301" class="Bound">eq₂</a><a id="6342" class="Symbol">)</a>
<a id="6360" class="Symbol">}</a>
<a id="6374" class="Symbol">}</a>
<a id="6384" class="Symbol">}</a>
<a id="6387" class="Comment">-- extra properties of &quot;up-to-iso&quot;</a>
<a id="6422" class="Keyword">module</a> <a id="IsoPb"></a><a id="6429" href="Categories.Diagram.Pullback.Properties.html#6429" class="Module">IsoPb</a> <a id="6435" class="Symbol">{</a><a id="6436" href="Categories.Diagram.Pullback.Properties.html#6436" class="Bound">X</a> <a id="6438" href="Categories.Diagram.Pullback.Properties.html#6438" class="Bound">Y</a> <a id="6440" href="Categories.Diagram.Pullback.Properties.html#6440" class="Bound">Z</a><a id="6441" class="Symbol">}</a> <a id="6443" class="Symbol">{</a><a id="6444" href="Categories.Diagram.Pullback.Properties.html#6444" class="Bound">f</a> <a id="6446" class="Symbol">:</a> <a id="6448" href="Categories.Diagram.Pullback.Properties.html#6436" class="Bound">X</a> <a id="6450" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="6452" href="Categories.Diagram.Pullback.Properties.html#6440" class="Bound">Z</a><a id="6453" class="Symbol">}</a> <a id="6455" class="Symbol">{</a><a id="6456" href="Categories.Diagram.Pullback.Properties.html#6456" class="Bound">g</a> <a id="6458" class="Symbol">:</a> <a id="6460" href="Categories.Diagram.Pullback.Properties.html#6438" class="Bound">Y</a> <a id="6462" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="6464" href="Categories.Diagram.Pullback.Properties.html#6440" class="Bound">Z</a><a id="6465" class="Symbol">}</a> <a id="6467" class="Symbol">(</a><a id="6468" href="Categories.Diagram.Pullback.Properties.html#6468" class="Bound">pull₀</a> <a id="6474" href="Categories.Diagram.Pullback.Properties.html#6474" class="Bound">pull₁</a> <a id="6480" class="Symbol">:</a> <a id="6482" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="6491" href="Categories.Diagram.Pullback.Properties.html#6444" class="Bound">f</a> <a id="6493" href="Categories.Diagram.Pullback.Properties.html#6456" class="Bound">g</a><a id="6494" class="Symbol">)</a> <a id="6496" class="Keyword">where</a>
<a id="6504" class="Keyword">open</a> <a id="6509" href="Categories.Diagram.Pullback.html#2029" class="Module">Pullback</a> <a id="6518" class="Keyword">using</a> <a id="6524" class="Symbol">(</a><a id="6525" href="Categories.Diagram.Pullback.html#2099" class="Field">P</a><a id="6526" class="Symbol">;</a> <a id="6528" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a><a id="6530" class="Symbol">;</a> <a id="6532" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a><a id="6534" class="Symbol">;</a> <a id="6536" href="Categories.Diagram.Pullback.html#1081" class="Function">p₁∘universal≈h₁</a><a id="6551" class="Symbol">;</a> <a id="6553" href="Categories.Diagram.Pullback.html#1179" class="Function">p₂∘universal≈h₂</a><a id="6568" class="Symbol">;</a> <a id="6570" href="Categories.Diagram.Pullback.html#854" class="Function">commute</a><a id="6577" class="Symbol">;</a> <a id="6579" href="Categories.Diagram.Pullback.html#886" class="Function">universal</a><a id="6588" class="Symbol">)</a>
<a id="IsoPb.P₀≅P₁"></a><a id="6593" href="Categories.Diagram.Pullback.Properties.html#6593" class="Function">P₀≅P₁</a> <a id="6599" class="Symbol">:</a> <a id="6601" href="Categories.Diagram.Pullback.html#2099" class="Field">P</a> <a id="6603" href="Categories.Diagram.Pullback.Properties.html#6468" class="Bound">pull₀</a> <a id="6609" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="6611" href="Categories.Diagram.Pullback.html#2099" class="Field">P</a> <a id="6613" href="Categories.Diagram.Pullback.Properties.html#6474" class="Bound">pull₁</a>
<a id="6621" href="Categories.Diagram.Pullback.Properties.html#6593" class="Function">P₀≅P₁</a> <a id="6627" class="Symbol">=</a> <a id="6629" href="Categories.Diagram.Pullback.html#2216" class="Function">up-to-iso</a> <a id="6639" href="Categories.Diagram.Pullback.Properties.html#6468" class="Bound">pull₀</a> <a id="6645" href="Categories.Diagram.Pullback.Properties.html#6474" class="Bound">pull₁</a>
<a id="IsoPb.P₀⇒P₁"></a><a id="6654" href="Categories.Diagram.Pullback.Properties.html#6654" class="Function">P₀⇒P₁</a> <a id="6660" class="Symbol">:</a> <a id="6662" href="Categories.Diagram.Pullback.html#2099" class="Field">P</a> <a id="6664" href="Categories.Diagram.Pullback.Properties.html#6468" class="Bound">pull₀</a> <a id="6670" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="6672" href="Categories.Diagram.Pullback.html#2099" class="Field">P</a> <a id="6674" href="Categories.Diagram.Pullback.Properties.html#6474" class="Bound">pull₁</a>
<a id="6682" href="Categories.Diagram.Pullback.Properties.html#6654" class="Function">P₀⇒P₁</a> <a id="6688" class="Symbol">=</a> <a id="6690" href="Categories.Morphism.html#2006" class="Field">_≅_.from</a> <a id="6699" href="Categories.Diagram.Pullback.Properties.html#6593" class="Function">P₀≅P₁</a>
<a id="IsoPb.p₁-≈"></a><a id="6708" href="Categories.Diagram.Pullback.Properties.html#6708" class="Function">p₁-≈</a> <a id="6713" class="Symbol">:</a> <a id="6715" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="6718" href="Categories.Diagram.Pullback.Properties.html#6474" class="Bound">pull₁</a> <a id="6724" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="6726" href="Categories.Diagram.Pullback.Properties.html#6654" class="Function">P₀⇒P₁</a> <a id="6732" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="6734" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="6737" href="Categories.Diagram.Pullback.Properties.html#6468" class="Bound">pull₀</a>
<a id="6745" href="Categories.Diagram.Pullback.Properties.html#6708" class="Function">p₁-≈</a> <a id="6750" class="Symbol">=</a> <a id="6752" href="Categories.Diagram.Pullback.html#1081" class="Function">p₁∘universal≈h₁</a> <a id="6768" href="Categories.Diagram.Pullback.Properties.html#6474" class="Bound">pull₁</a> <a id="6774" class="Symbol">{</a><a id="6775" class="Argument">eq</a> <a id="6778" class="Symbol">=</a> <a id="6780" href="Categories.Diagram.Pullback.html#854" class="Function">commute</a> <a id="6788" href="Categories.Diagram.Pullback.Properties.html#6468" class="Bound">pull₀</a><a id="6793" class="Symbol">}</a>
<a id="IsoPb.p₂-≈"></a><a id="6798" href="Categories.Diagram.Pullback.Properties.html#6798" class="Function">p₂-≈</a> <a id="6803" class="Symbol">:</a> <a id="6805" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="6808" href="Categories.Diagram.Pullback.Properties.html#6474" class="Bound">pull₁</a> <a id="6814" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="6816" href="Categories.Diagram.Pullback.Properties.html#6654" class="Function">P₀⇒P₁</a> <a id="6822" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="6824" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="6827" href="Categories.Diagram.Pullback.Properties.html#6468" class="Bound">pull₀</a>
<a id="6835" href="Categories.Diagram.Pullback.Properties.html#6798" class="Function">p₂-≈</a> <a id="6840" class="Symbol">=</a> <a id="6842" href="Categories.Diagram.Pullback.html#1179" class="Function">p₂∘universal≈h₂</a> <a id="6858" href="Categories.Diagram.Pullback.Properties.html#6474" class="Bound">pull₁</a> <a id="6864" class="Symbol">{</a><a id="6865" class="Argument">eq</a> <a id="6868" class="Symbol">=</a> <a id="6870" href="Categories.Diagram.Pullback.html#854" class="Function">commute</a> <a id="6878" href="Categories.Diagram.Pullback.Properties.html#6468" class="Bound">pull₀</a><a id="6883" class="Symbol">}</a>
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