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<html><head><meta charset="utf-8"><title>Categories.Functor.Limits</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="36" class="Keyword">module</a> <a id="43" href="Categories.Functor.Limits.html" class="Module">Categories.Functor.Limits</a> <a id="69" class="Keyword">where</a>
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<a id="76" class="Keyword">open</a> <a id="81" class="Keyword">import</a> <a id="88" href="Level.html" class="Module">Level</a>
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<a id="95" class="Keyword">open</a> <a id="100" class="Keyword">import</a> <a id="107" href="Categories.Category.html" class="Module">Categories.Category</a>
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<a id="127" class="Keyword">open</a> <a id="132" class="Keyword">import</a> <a id="139" href="Categories.Functor.html" class="Module">Categories.Functor</a>
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<a id="158" class="Keyword">open</a> <a id="163" class="Keyword">import</a> <a id="170" href="Categories.Functor.Properties.html" class="Module">Categories.Functor.Properties</a>
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<a id="200" class="Keyword">open</a> <a id="205" class="Keyword">import</a> <a id="212" href="Categories.Object.Terminal.html" class="Module">Categories.Object.Terminal</a>
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<a id="239" class="Keyword">open</a> <a id="244" class="Keyword">import</a> <a id="251" href="Categories.Object.Initial.html" class="Module">Categories.Object.Initial</a>
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<a id="278" class="Keyword">open</a> <a id="283" class="Keyword">import</a> <a id="290" href="Categories.Diagram.Limit.html" class="Module">Categories.Diagram.Limit</a>
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<a id="315" class="Keyword">open</a> <a id="320" class="Keyword">import</a> <a id="327" href="Categories.Diagram.Colimit.html" class="Module">Categories.Diagram.Colimit</a>
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<a id="354" class="Keyword">open</a> <a id="359" class="Keyword">import</a> <a id="366" href="Categories.Diagram.Cone.Properties.html" class="Module">Categories.Diagram.Cone.Properties</a>
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<a id="401" class="Keyword">open</a> <a id="406" class="Keyword">import</a> <a id="413" href="Categories.Diagram.Cocone.Properties.html" class="Module">Categories.Diagram.Cocone.Properties</a>
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<a id="451" class="Keyword">open</a> <a id="456" class="Keyword">import</a> <a id="463" href="Categories.Category.Construction.Cones.html" class="Module">Categories.Category.Construction.Cones</a>
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<a id="502" class="Keyword">open</a> <a id="507" class="Keyword">import</a> <a id="514" href="Categories.Category.Construction.Cocones.html" class="Module">Categories.Category.Construction.Cocones</a>
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<a id="556" class="Keyword">private</a>
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<a id="566" class="Keyword">variable</a>
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<a id="579" href="Categories.Functor.Limits.html#579" class="Generalizable">o</a> <a id="581" href="Categories.Functor.Limits.html#581" class="Generalizable">ℓ</a> <a id="583" href="Categories.Functor.Limits.html#583" class="Generalizable">e</a> <a id="585" class="Symbol">:</a> <a id="587" href="Agda.Primitive.html#742" class="Postulate">Level</a>
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<a id="597" href="Categories.Functor.Limits.html#597" class="Generalizable">𝒞</a> <a id="599" href="Categories.Functor.Limits.html#599" class="Generalizable">𝒟</a> <a id="601" href="Categories.Functor.Limits.html#601" class="Generalizable">ℐ</a> <a id="603" class="Symbol">:</a> <a id="605" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="614" href="Categories.Functor.Limits.html#579" class="Generalizable">o</a> <a id="616" href="Categories.Functor.Limits.html#581" class="Generalizable">ℓ</a> <a id="618" href="Categories.Functor.Limits.html#583" class="Generalizable">e</a>
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<a id="621" class="Keyword">module</a> <a id="628" href="Categories.Functor.Limits.html#628" class="Module">_</a> <a id="630" class="Symbol">(</a><a id="631" href="Categories.Functor.Limits.html#631" class="Bound">F</a> <a id="633" class="Symbol">:</a> <a id="635" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="643" href="Categories.Functor.Limits.html#597" class="Generalizable">𝒞</a> <a id="645" href="Categories.Functor.Limits.html#599" class="Generalizable">𝒟</a><a id="646" class="Symbol">)</a> <a id="648" class="Symbol">{</a><a id="649" href="Categories.Functor.Limits.html#649" class="Bound">J</a> <a id="651" class="Symbol">:</a> <a id="653" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="661" href="Categories.Functor.Limits.html#601" class="Generalizable">ℐ</a> <a id="663" href="Categories.Functor.Limits.html#597" class="Generalizable">𝒞</a><a id="664" class="Symbol">}</a> <a id="666" class="Keyword">where</a>
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<a id="675" href="Categories.Functor.Limits.html#675" class="Function">PreservesLimit</a> <a id="690" class="Symbol">:</a> <a id="692" class="Symbol">(</a><a id="693" href="Categories.Functor.Limits.html#693" class="Bound">L</a> <a id="695" class="Symbol">:</a> <a id="697" href="Categories.Diagram.Limit.html#934" class="Record">Limit</a> <a id="703" href="Categories.Functor.Limits.html#649" class="Bound">J</a><a id="704" class="Symbol">)</a> <a id="706" class="Symbol">→</a> <a id="708" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="712" class="Symbol">_</a>
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<a id="716" href="Categories.Functor.Limits.html#675" class="Function">PreservesLimit</a> <a id="731" href="Categories.Functor.Limits.html#731" class="Bound">L</a> <a id="733" class="Symbol">=</a> <a id="735" href="Categories.Object.Terminal.html#520" class="Record">IsTerminal</a> <a id="746" class="Symbol">(</a><a id="747" href="Categories.Category.Construction.Cones.html#632" class="Function">Cones</a> <a id="753" class="Symbol">(</a><a id="754" href="Categories.Functor.Limits.html#631" class="Bound">F</a> <a id="756" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="759" href="Categories.Functor.Limits.html#649" class="Bound">J</a><a id="760" class="Symbol">))</a> <a id="763" class="Symbol">(</a><a id="764" href="Categories.Diagram.Cone.Properties.html#633" class="Function">F-map-Coneˡ</a> <a id="776" href="Categories.Functor.Limits.html#631" class="Bound">F</a> <a id="778" href="Categories.Diagram.Limit.html#1097" class="Function">limit</a><a id="783" class="Symbol">)</a>
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<a id="789" class="Keyword">where</a> <a id="795" class="Keyword">open</a> <a id="800" href="Categories.Diagram.Limit.html#934" class="Module">Limit</a> <a id="806" href="Categories.Functor.Limits.html#731" class="Bound">L</a>
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<a id="811" href="Categories.Functor.Limits.html#811" class="Function">PreservesColimit</a> <a id="828" class="Symbol">:</a> <a id="830" class="Symbol">(</a><a id="831" href="Categories.Functor.Limits.html#831" class="Bound">L</a> <a id="833" class="Symbol">:</a> <a id="835" href="Categories.Diagram.Colimit.html#961" class="Record">Colimit</a> <a id="843" href="Categories.Functor.Limits.html#649" class="Bound">J</a><a id="844" class="Symbol">)</a> <a id="846" class="Symbol">→</a> <a id="848" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="852" class="Symbol">_</a>
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<a id="856" href="Categories.Functor.Limits.html#811" class="Function">PreservesColimit</a> <a id="873" href="Categories.Functor.Limits.html#873" class="Bound">L</a> <a id="875" class="Symbol">=</a> <a id="877" href="Categories.Object.Initial.html#422" class="Record">IsInitial</a> <a id="887" class="Symbol">(</a><a id="888" href="Categories.Category.Construction.Cocones.html#731" class="Function">Cocones</a> <a id="896" class="Symbol">(</a><a id="897" href="Categories.Functor.Limits.html#631" class="Bound">F</a> <a id="899" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="902" href="Categories.Functor.Limits.html#649" class="Bound">J</a><a id="903" class="Symbol">))</a> <a id="906" class="Symbol">(</a><a id="907" href="Categories.Diagram.Cocone.Properties.html#724" class="Function">F-map-Coconeˡ</a> <a id="921" href="Categories.Functor.Limits.html#631" class="Bound">F</a> <a id="923" href="Categories.Diagram.Colimit.html#1122" class="Function">colimit</a><a id="930" class="Symbol">)</a>
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<a id="936" class="Keyword">where</a> <a id="942" class="Keyword">open</a> <a id="947" href="Categories.Diagram.Colimit.html#961" class="Module">Colimit</a> <a id="955" href="Categories.Functor.Limits.html#873" class="Bound">L</a>
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<a id="960" href="Categories.Functor.Limits.html#960" class="Function">ReflectsLimits</a> <a id="975" class="Symbol">:</a> <a id="977" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="981" class="Symbol">_</a>
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<a id="985" href="Categories.Functor.Limits.html#960" class="Function">ReflectsLimits</a> <a id="1000" class="Symbol">=</a> <a id="1002" class="Symbol">∀</a> <a id="1004" class="Symbol">(</a><a id="1005" href="Categories.Functor.Limits.html#1005" class="Bound">K</a> <a id="1007" class="Symbol">:</a> <a id="1009" href="Categories.Diagram.Cone.html#526" class="Record">Cone</a> <a id="1014" href="Categories.Functor.Limits.html#649" class="Bound">J</a><a id="1015" class="Symbol">)</a> <a id="1017" class="Symbol">→</a> <a id="1019" href="Categories.Object.Terminal.html#520" class="Record">IsTerminal</a> <a id="1030" class="Symbol">(</a><a id="1031" href="Categories.Category.Construction.Cones.html#632" class="Function">Cones</a> <a id="1037" class="Symbol">(</a><a id="1038" href="Categories.Functor.Limits.html#631" class="Bound">F</a> <a id="1040" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="1043" href="Categories.Functor.Limits.html#649" class="Bound">J</a><a id="1044" class="Symbol">))</a> <a id="1047" class="Symbol">(</a><a id="1048" href="Categories.Diagram.Cone.Properties.html#633" class="Function">F-map-Coneˡ</a> <a id="1060" href="Categories.Functor.Limits.html#631" class="Bound">F</a> <a id="1062" href="Categories.Functor.Limits.html#1005" class="Bound">K</a><a id="1063" class="Symbol">)</a> <a id="1065" class="Symbol">→</a> <a id="1067" href="Categories.Object.Terminal.html#520" class="Record">IsTerminal</a> <a id="1078" class="Symbol">(</a><a id="1079" href="Categories.Category.Construction.Cones.html#632" class="Function">Cones</a> <a id="1085" href="Categories.Functor.Limits.html#649" class="Bound">J</a><a id="1086" class="Symbol">)</a> <a id="1088" href="Categories.Functor.Limits.html#1005" class="Bound">K</a>
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<a id="1093" href="Categories.Functor.Limits.html#1093" class="Function">ReflectsColimits</a> <a id="1110" class="Symbol">:</a> <a id="1112" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1116" class="Symbol">_</a>
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<a id="1120" href="Categories.Functor.Limits.html#1093" class="Function">ReflectsColimits</a> <a id="1137" class="Symbol">=</a> <a id="1139" class="Symbol">∀</a> <a id="1141" class="Symbol">(</a><a id="1142" href="Categories.Functor.Limits.html#1142" class="Bound">K</a> <a id="1144" class="Symbol">:</a> <a id="1146" href="Categories.Diagram.Cocone.html#534" class="Record">Cocone</a> <a id="1153" href="Categories.Functor.Limits.html#649" class="Bound">J</a><a id="1154" class="Symbol">)</a> <a id="1156" class="Symbol">→</a> <a id="1158" href="Categories.Object.Initial.html#422" class="Record">IsInitial</a> <a id="1168" class="Symbol">(</a><a id="1169" href="Categories.Category.Construction.Cocones.html#731" class="Function">Cocones</a> <a id="1177" class="Symbol">(</a><a id="1178" href="Categories.Functor.Limits.html#631" class="Bound">F</a> <a id="1180" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="1183" href="Categories.Functor.Limits.html#649" class="Bound">J</a><a id="1184" class="Symbol">))</a> <a id="1187" class="Symbol">(</a><a id="1188" href="Categories.Diagram.Cocone.Properties.html#724" class="Function">F-map-Coconeˡ</a> <a id="1202" href="Categories.Functor.Limits.html#631" class="Bound">F</a> <a id="1204" href="Categories.Functor.Limits.html#1142" class="Bound">K</a><a id="1205" class="Symbol">)</a> <a id="1207" class="Symbol">→</a> <a id="1209" href="Categories.Object.Initial.html#422" class="Record">IsInitial</a> <a id="1219" class="Symbol">(</a><a id="1220" href="Categories.Category.Construction.Cocones.html#731" class="Function">Cocones</a> <a id="1228" href="Categories.Functor.Limits.html#649" class="Bound">J</a><a id="1229" class="Symbol">)</a> <a id="1231" href="Categories.Functor.Limits.html#1142" class="Bound">K</a>
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<a id="1234" class="Comment">-- record CreatesLimits : Set (o ⊔ ℓ ⊔ e ⊔ o′ ⊔ ℓ′ ⊔ e′ ⊔ o″ ⊔ ℓ″) where</a>
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<a id="1309" class="Comment">-- field</a>
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<a id="1322" class="Comment">-- preserves-limits : PreservesLimit</a>
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<a id="1365" class="Comment">-- reflects-limits : ReflectsLimits</a>
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<a id="1409" class="Comment">-- record CreatesColimits : Set (o ⊔ ℓ ⊔ e ⊔ o′ ⊔ ℓ′ ⊔ e′ ⊔ o″ ⊔ ℓ″) where</a>
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<a id="1486" class="Comment">-- field</a>
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<a id="1499" class="Comment">-- preserves-colimits : PreservesColimit</a>
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<a id="1546" class="Comment">-- reflects-colimits : ReflectsColimits</a>
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<a id="Continuous"></a><a id="1594" href="Categories.Functor.Limits.html#1594" class="Function">Continuous</a> <a id="1605" class="Symbol">:</a> <a id="1607" class="Symbol">∀</a> <a id="1609" href="Categories.Functor.Limits.html#1609" class="Bound">o</a> <a id="1611" href="Categories.Functor.Limits.html#1611" class="Bound">ℓ</a> <a id="1613" href="Categories.Functor.Limits.html#1613" class="Bound">e</a> <a id="1615" class="Symbol">→</a> <a id="1617" class="Symbol">(</a><a id="1618" href="Categories.Functor.Limits.html#1618" class="Bound">F</a> <a id="1620" class="Symbol">:</a> <a id="1622" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="1630" href="Categories.Functor.Limits.html#597" class="Generalizable">𝒞</a> <a id="1632" href="Categories.Functor.Limits.html#599" class="Generalizable">𝒟</a><a id="1633" class="Symbol">)</a> <a id="1635" class="Symbol">→</a> <a id="1637" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1641" class="Symbol">_</a>
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<a id="1643" href="Categories.Functor.Limits.html#1594" class="Function">Continuous</a> <a id="1654" class="Symbol">{</a><a id="1655" class="Argument">𝒞</a> <a id="1657" class="Symbol">=</a> <a id="1659" href="Categories.Functor.Limits.html#1659" class="Bound">𝒞</a><a id="1660" class="Symbol">}</a> <a id="1662" href="Categories.Functor.Limits.html#1662" class="Bound">o</a> <a id="1664" href="Categories.Functor.Limits.html#1664" class="Bound">ℓ</a> <a id="1666" href="Categories.Functor.Limits.html#1666" class="Bound">e</a> <a id="1668" href="Categories.Functor.Limits.html#1668" class="Bound">F</a> <a id="1670" class="Symbol">=</a> <a id="1672" class="Symbol">∀</a> <a id="1674" class="Symbol">{</a><a id="1675" href="Categories.Functor.Limits.html#1675" class="Bound">𝒥</a> <a id="1677" class="Symbol">:</a> <a id="1679" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="1688" href="Categories.Functor.Limits.html#1662" class="Bound">o</a> <a id="1690" href="Categories.Functor.Limits.html#1664" class="Bound">ℓ</a> <a id="1692" href="Categories.Functor.Limits.html#1666" class="Bound">e</a><a id="1693" class="Symbol">}</a> <a id="1695" class="Symbol">{</a><a id="1696" href="Categories.Functor.Limits.html#1696" class="Bound">J</a> <a id="1698" class="Symbol">:</a> <a id="1700" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="1708" href="Categories.Functor.Limits.html#1675" class="Bound">𝒥</a> <a id="1710" href="Categories.Functor.Limits.html#1659" class="Bound">𝒞</a><a id="1711" class="Symbol">}</a> <a id="1713" class="Symbol">(</a><a id="1714" href="Categories.Functor.Limits.html#1714" class="Bound">L</a> <a id="1716" class="Symbol">:</a> <a id="1718" href="Categories.Diagram.Limit.html#934" class="Record">Limit</a> <a id="1724" href="Categories.Functor.Limits.html#1696" class="Bound">J</a><a id="1725" class="Symbol">)</a> <a id="1727" class="Symbol">→</a> <a id="1729" href="Categories.Functor.Limits.html#675" class="Function">PreservesLimit</a> <a id="1744" href="Categories.Functor.Limits.html#1668" class="Bound">F</a> <a id="1746" href="Categories.Functor.Limits.html#1714" class="Bound">L</a>
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<a id="Cocontinuous"></a><a id="1749" href="Categories.Functor.Limits.html#1749" class="Function">Cocontinuous</a> <a id="1762" class="Symbol">:</a> <a id="1764" class="Symbol">∀</a> <a id="1766" href="Categories.Functor.Limits.html#1766" class="Bound">o</a> <a id="1768" href="Categories.Functor.Limits.html#1768" class="Bound">ℓ</a> <a id="1770" href="Categories.Functor.Limits.html#1770" class="Bound">e</a> <a id="1772" class="Symbol">→</a> <a id="1774" class="Symbol">(</a><a id="1775" href="Categories.Functor.Limits.html#1775" class="Bound">F</a> <a id="1777" class="Symbol">:</a> <a id="1779" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="1787" href="Categories.Functor.Limits.html#597" class="Generalizable">𝒞</a> <a id="1789" href="Categories.Functor.Limits.html#599" class="Generalizable">𝒟</a><a id="1790" class="Symbol">)</a> <a id="1792" class="Symbol">→</a> <a id="1794" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1798" class="Symbol">_</a>
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<a id="1800" href="Categories.Functor.Limits.html#1749" class="Function">Cocontinuous</a> <a id="1813" class="Symbol">{</a><a id="1814" class="Argument">𝒞</a> <a id="1816" class="Symbol">=</a> <a id="1818" href="Categories.Functor.Limits.html#1818" class="Bound">𝒞</a><a id="1819" class="Symbol">}</a> <a id="1821" href="Categories.Functor.Limits.html#1821" class="Bound">o</a> <a id="1823" href="Categories.Functor.Limits.html#1823" class="Bound">ℓ</a> <a id="1825" href="Categories.Functor.Limits.html#1825" class="Bound">e</a> <a id="1827" href="Categories.Functor.Limits.html#1827" class="Bound">F</a> <a id="1829" class="Symbol">=</a> <a id="1831" class="Symbol">∀</a> <a id="1833" class="Symbol">{</a><a id="1834" href="Categories.Functor.Limits.html#1834" class="Bound">𝒥</a> <a id="1836" class="Symbol">:</a> <a id="1838" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="1847" href="Categories.Functor.Limits.html#1821" class="Bound">o</a> <a id="1849" href="Categories.Functor.Limits.html#1823" class="Bound">ℓ</a> <a id="1851" href="Categories.Functor.Limits.html#1825" class="Bound">e</a><a id="1852" class="Symbol">}</a> <a id="1854" class="Symbol">{</a><a id="1855" href="Categories.Functor.Limits.html#1855" class="Bound">J</a> <a id="1857" class="Symbol">:</a> <a id="1859" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="1867" href="Categories.Functor.Limits.html#1834" class="Bound">𝒥</a> <a id="1869" href="Categories.Functor.Limits.html#1818" class="Bound">𝒞</a><a id="1870" class="Symbol">}</a> <a id="1872" class="Symbol">(</a><a id="1873" href="Categories.Functor.Limits.html#1873" class="Bound">L</a> <a id="1875" class="Symbol">:</a> <a id="1877" href="Categories.Diagram.Colimit.html#961" class="Record">Colimit</a> <a id="1885" href="Categories.Functor.Limits.html#1855" class="Bound">J</a><a id="1886" class="Symbol">)</a> <a id="1888" class="Symbol">→</a> <a id="1890" href="Categories.Functor.Limits.html#811" class="Function">PreservesColimit</a> <a id="1907" href="Categories.Functor.Limits.html#1827" class="Bound">F</a> <a id="1909" href="Categories.Functor.Limits.html#1873" class="Bound">L</a>
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