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114 lines
50 KiB
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<html><head><meta charset="utf-8"><title>Categories.NaturalTransformation.Properties</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Symbol">{-#</a> <a id="5" class="Keyword">OPTIONS</a> <a id="13" class="Pragma">--without-K</a> <a id="25" class="Pragma">--safe</a> <a id="32" class="Symbol">#-}</a>
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<a id="37" class="Keyword">module</a> <a id="44" href="Categories.NaturalTransformation.Properties.html" class="Module">Categories.NaturalTransformation.Properties</a> <a id="88" class="Keyword">where</a>
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<a id="95" class="Keyword">open</a> <a id="100" class="Keyword">import</a> <a id="107" href="Level.html" class="Module">Level</a>
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<a id="113" class="Keyword">open</a> <a id="118" class="Keyword">import</a> <a id="125" href="Data.Product.html" class="Module">Data.Product</a> <a id="138" class="Keyword">using</a> <a id="144" class="Symbol">(</a><a id="145" href="Agda.Builtin.Sigma.html#165" class="Record">Σ</a><a id="146" class="Symbol">;</a> <a id="148" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="151" class="Symbol">)</a>
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<a id="153" class="Keyword">open</a> <a id="158" class="Keyword">import</a> <a id="165" href="Function.Bundles.html" class="Module">Function.Bundles</a> <a id="182" class="Keyword">using</a> <a id="188" class="Symbol">(</a><a id="189" href="Function.Bundles.html#2043" class="Record">Func</a><a id="193" class="Symbol">)</a>
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<a id="196" class="Keyword">open</a> <a id="201" class="Keyword">import</a> <a id="208" href="Categories.Category.html" class="Module">Categories.Category</a>
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<a id="228" class="Keyword">open</a> <a id="233" class="Keyword">import</a> <a id="240" href="Categories.Category.Product.html" class="Module">Categories.Category.Product</a>
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<a id="268" class="Keyword">open</a> <a id="273" class="Keyword">import</a> <a id="280" href="Categories.Category.Instance.Setoids.html" class="Module">Categories.Category.Instance.Setoids</a>
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<a id="317" class="Keyword">open</a> <a id="322" class="Keyword">import</a> <a id="329" href="Categories.Functor.html" class="Module">Categories.Functor</a>
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<a id="348" class="Keyword">open</a> <a id="353" class="Keyword">import</a> <a id="360" href="Categories.Functor.Construction.Constant.html" class="Module">Categories.Functor.Construction.Constant</a>
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<a id="401" class="Keyword">open</a> <a id="406" class="Keyword">import</a> <a id="413" href="Categories.Functor.Construction.LiftSetoids.html" class="Module">Categories.Functor.Construction.LiftSetoids</a>
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<a id="457" class="Keyword">open</a> <a id="462" class="Keyword">import</a> <a id="469" href="Categories.Functor.Bifunctor.html" class="Module">Categories.Functor.Bifunctor</a>
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<a id="498" class="Keyword">open</a> <a id="503" class="Keyword">import</a> <a id="510" href="Categories.NaturalTransformation.html" class="Module">Categories.NaturalTransformation</a> <a id="543" class="Keyword">renaming</a> <a id="552" class="Symbol">(</a><a id="553" href="Categories.NaturalTransformation.Core.html#2132" class="Function">id</a> <a id="556" class="Symbol">to</a> <a id="559" class="Function">idN</a><a id="562" class="Symbol">)</a>
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<a id="564" class="Keyword">open</a> <a id="569" class="Keyword">import</a> <a id="576" href="Categories.NaturalTransformation.NaturalIsomorphism.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism</a>
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<a id="628" class="Keyword">open</a> <a id="633" class="Keyword">import</a> <a id="640" href="Categories.NaturalTransformation.Dinatural.html" class="Module">Categories.NaturalTransformation.Dinatural</a> <a id="683" class="Keyword">hiding</a> <a id="690" class="Symbol">(</a><a id="691" href="Categories.NaturalTransformation.Dinatural.html#6011" class="Function Operator">_≃_</a><a id="694" class="Symbol">)</a>
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<a id="697" class="Keyword">import</a> <a id="704" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="734" class="Symbol">as</a> <a id="737" class="Module">MR</a>
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<a id="741" class="Keyword">private</a>
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<a id="751" class="Keyword">variable</a>
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<a id="764" href="Categories.NaturalTransformation.Properties.html#764" class="Generalizable">o</a> <a id="766" href="Categories.NaturalTransformation.Properties.html#766" class="Generalizable">ℓ</a> <a id="768" href="Categories.NaturalTransformation.Properties.html#768" class="Generalizable">ℓ′</a> <a id="771" href="Categories.NaturalTransformation.Properties.html#771" class="Generalizable">e</a> <a id="773" class="Symbol">:</a> <a id="775" href="Agda.Primitive.html#742" class="Postulate">Level</a>
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<a id="785" href="Categories.NaturalTransformation.Properties.html#785" class="Generalizable">C</a> <a id="787" href="Categories.NaturalTransformation.Properties.html#787" class="Generalizable">D</a> <a id="789" href="Categories.NaturalTransformation.Properties.html#789" class="Generalizable">E</a> <a id="791" class="Symbol">:</a> <a id="793" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="802" href="Categories.NaturalTransformation.Properties.html#764" class="Generalizable">o</a> <a id="804" href="Categories.NaturalTransformation.Properties.html#766" class="Generalizable">ℓ</a> <a id="806" href="Categories.NaturalTransformation.Properties.html#771" class="Generalizable">e</a>
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<a id="809" class="Keyword">module</a> <a id="816" href="Categories.NaturalTransformation.Properties.html#816" class="Module">_</a> <a id="818" class="Symbol">{</a><a id="819" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">F</a> <a id="821" href="Categories.NaturalTransformation.Properties.html#821" class="Bound">G</a> <a id="823" class="Symbol">:</a> <a id="825" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="833" href="Categories.NaturalTransformation.Properties.html#785" class="Generalizable">C</a> <a id="835" href="Categories.NaturalTransformation.Properties.html#787" class="Generalizable">D</a><a id="836" class="Symbol">}</a> <a id="838" class="Keyword">where</a>
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<a id="846" class="Keyword">private</a>
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<a id="858" class="Keyword">module</a> <a id="865" href="Categories.NaturalTransformation.Properties.html#865" class="Module">C</a> <a id="867" class="Symbol">=</a> <a id="869" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="878" href="Categories.NaturalTransformation.Properties.html#833" class="Bound">C</a>
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<a id="884" class="Keyword">module</a> <a id="891" href="Categories.NaturalTransformation.Properties.html#891" class="Module">D</a> <a id="893" class="Symbol">=</a> <a id="895" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="904" href="Categories.NaturalTransformation.Properties.html#835" class="Bound">D</a>
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<a id="910" class="Keyword">open</a> <a id="915" href="Categories.NaturalTransformation.Properties.html#891" class="Module">D</a>
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<a id="921" class="Keyword">open</a> <a id="926" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="929" href="Categories.NaturalTransformation.Properties.html#835" class="Bound">D</a>
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<a id="935" class="Keyword">open</a> <a id="940" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
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<a id="957" class="Keyword">open</a> <a id="962" href="Categories.Functor.Core.html#248" class="Module">Functor</a>
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<a id="975" href="Categories.NaturalTransformation.Properties.html#975" class="Function">F′</a> <a id="978" class="Symbol">:</a> <a id="980" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="990" href="Categories.Category.Core.html#3132" class="Function">C.op</a> <a id="995" href="Categories.NaturalTransformation.Properties.html#833" class="Bound">C</a> <a id="997" href="Categories.NaturalTransformation.Properties.html#835" class="Bound">D</a>
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<a id="1003" href="Categories.NaturalTransformation.Properties.html#975" class="Function">F′</a> <a id="1006" class="Symbol">=</a> <a id="1008" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">F</a> <a id="1010" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="1013" href="Categories.Category.Product.html#6067" class="Function">πʳ</a>
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<a id="1021" href="Categories.NaturalTransformation.Properties.html#1021" class="Function">G′</a> <a id="1024" class="Symbol">:</a> <a id="1026" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="1036" href="Categories.Category.Core.html#3132" class="Function">C.op</a> <a id="1041" href="Categories.NaturalTransformation.Properties.html#833" class="Bound">C</a> <a id="1043" href="Categories.NaturalTransformation.Properties.html#835" class="Bound">D</a>
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<a id="1049" href="Categories.NaturalTransformation.Properties.html#1021" class="Function">G′</a> <a id="1052" class="Symbol">=</a> <a id="1054" href="Categories.NaturalTransformation.Properties.html#821" class="Bound">G</a> <a id="1056" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="1059" href="Categories.Category.Product.html#6067" class="Function">πʳ</a>
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<a id="1065" href="Categories.NaturalTransformation.Properties.html#1065" class="Function">NT⇒Dinatural</a> <a id="1078" class="Symbol">:</a> <a id="1080" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="1102" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">F</a> <a id="1104" href="Categories.NaturalTransformation.Properties.html#821" class="Bound">G</a> <a id="1106" class="Symbol">→</a> <a id="1108" href="Categories.NaturalTransformation.Dinatural.html#578" class="Record">DinaturalTransformation</a> <a id="1132" href="Categories.NaturalTransformation.Properties.html#975" class="Function">F′</a> <a id="1135" href="Categories.NaturalTransformation.Properties.html#1021" class="Function">G′</a>
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<a id="1140" href="Categories.NaturalTransformation.Properties.html#1065" class="Function">NT⇒Dinatural</a> <a id="1153" href="Categories.NaturalTransformation.Properties.html#1153" class="Bound">β</a> <a id="1155" class="Symbol">=</a> <a id="1157" href="Categories.NaturalTransformation.Dinatural.html#2924" class="Function">dtHelper</a> <a id="1166" class="Keyword">record</a>
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<a id="1177" class="Symbol">{</a> <a id="1179" href="Categories.NaturalTransformation.Dinatural.html#2434" class="Field">α</a> <a id="1187" class="Symbol">=</a> <a id="1189" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a>
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<a id="1195" class="Symbol">;</a> <a id="1197" href="Categories.NaturalTransformation.Dinatural.html#2491" class="Field">commute</a> <a id="1205" class="Symbol">=</a> <a id="1207" class="Symbol">λ</a> <a id="1209" href="Categories.NaturalTransformation.Properties.html#1209" class="Bound">f</a> <a id="1211" class="Symbol">→</a> <a id="1213" href="Categories.Category.Core.html#1706" class="Function">∘-resp-≈ʳ</a> <a id="1223" class="Symbol">(</a><a id="1224" href="Categories.Morphism.Reasoning.Core.html#2786" class="Function">elimʳ</a> <a id="1230" class="Symbol">(</a><a id="1231" href="Categories.Functor.Core.html#511" class="Field">identity</a> <a id="1240" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">F</a><a id="1241" class="Symbol">))</a> <a id="1244" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1246" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="1248" class="Symbol">(</a><a id="1249" href="Categories.NaturalTransformation.Core.html#827" class="Field">commute</a> <a id="1257" href="Categories.NaturalTransformation.Properties.html#1209" class="Bound">f</a><a id="1258" class="Symbol">)</a> <a id="1260" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1262" href="Categories.Morphism.Reasoning.Core.html#3063" class="Function">introˡ</a> <a id="1269" class="Symbol">(</a><a id="1270" href="Categories.Functor.Core.html#511" class="Field">identity</a> <a id="1279" href="Categories.NaturalTransformation.Properties.html#821" class="Bound">G</a><a id="1280" class="Symbol">)</a>
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<a id="1286" class="Symbol">}</a>
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<a id="1292" class="Keyword">where</a> <a id="1298" class="Keyword">open</a> <a id="1303" href="Categories.NaturalTransformation.Core.html#466" class="Module">NaturalTransformation</a> <a id="1325" href="Categories.NaturalTransformation.Properties.html#1153" class="Bound">β</a>
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<a id="1330" href="Categories.NaturalTransformation.Properties.html#1330" class="Function">Dinatural⇒NT</a> <a id="1343" class="Symbol">:</a> <a id="1345" href="Categories.NaturalTransformation.Dinatural.html#578" class="Record">DinaturalTransformation</a> <a id="1369" href="Categories.NaturalTransformation.Properties.html#975" class="Function">F′</a> <a id="1372" href="Categories.NaturalTransformation.Properties.html#1021" class="Function">G′</a> <a id="1375" class="Symbol">→</a> <a id="1377" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="1399" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">F</a> <a id="1401" href="Categories.NaturalTransformation.Properties.html#821" class="Bound">G</a>
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<a id="1405" href="Categories.NaturalTransformation.Properties.html#1330" class="Function">Dinatural⇒NT</a> <a id="1418" href="Categories.NaturalTransformation.Properties.html#1418" class="Bound">θ</a> <a id="1420" class="Symbol">=</a> <a id="1422" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="1431" class="Keyword">record</a>
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<a id="1442" class="Symbol">{</a> <a id="1444" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="1452" class="Symbol">=</a> <a id="1454" href="Categories.NaturalTransformation.Dinatural.html#851" class="Field">α</a>
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<a id="1460" class="Symbol">;</a> <a id="1462" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="1470" class="Symbol">=</a> <a id="1472" class="Symbol">λ</a> <a id="1474" href="Categories.NaturalTransformation.Properties.html#1474" class="Bound">f</a> <a id="1476" class="Symbol">→</a> <a id="1478" href="Categories.Morphism.Reasoning.Core.html#3063" class="Function">introˡ</a> <a id="1485" class="Symbol">(</a><a id="1486" href="Categories.Functor.Core.html#511" class="Field">identity</a> <a id="1495" href="Categories.NaturalTransformation.Properties.html#821" class="Bound">G</a><a id="1496" class="Symbol">)</a> <a id="1498" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1500" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="1502" class="Symbol">(</a><a id="1503" href="Categories.NaturalTransformation.Dinatural.html#908" class="Field">commute</a> <a id="1511" href="Categories.NaturalTransformation.Properties.html#1474" class="Bound">f</a><a id="1512" class="Symbol">)</a> <a id="1514" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1516" href="Categories.Category.Core.html#1706" class="Function">∘-resp-≈ʳ</a> <a id="1526" class="Symbol">(</a><a id="1527" href="Categories.Morphism.Reasoning.Core.html#2786" class="Function">elimʳ</a> <a id="1533" class="Symbol">(</a><a id="1534" href="Categories.Functor.Core.html#511" class="Field">identity</a> <a id="1543" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">F</a><a id="1544" class="Symbol">))</a>
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<a id="1551" class="Symbol">}</a>
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<a id="1557" class="Keyword">where</a> <a id="1563" class="Keyword">open</a> <a id="1568" href="Categories.NaturalTransformation.Dinatural.html#578" class="Module">DinaturalTransformation</a> <a id="1592" href="Categories.NaturalTransformation.Properties.html#1418" class="Bound">θ</a>
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<a id="1597" href="Categories.NaturalTransformation.Properties.html#1597" class="Function">replaceˡ</a> <a id="1606" class="Symbol">:</a> <a id="1608" class="Symbol">∀</a> <a id="1610" class="Symbol">{</a><a id="1611" href="Categories.NaturalTransformation.Properties.html#1611" class="Bound">F′</a><a id="1613" class="Symbol">}</a> <a id="1615" class="Symbol">→</a> <a id="1617" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="1639" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">F</a> <a id="1641" href="Categories.NaturalTransformation.Properties.html#821" class="Bound">G</a> <a id="1643" class="Symbol">→</a> <a id="1645" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">F</a> <a id="1647" href="Categories.NaturalTransformation.NaturalIsomorphism.html#3600" class="Function Operator">≃</a> <a id="1649" href="Categories.NaturalTransformation.Properties.html#1611" class="Bound">F′</a> <a id="1652" class="Symbol">→</a> <a id="1654" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="1676" href="Categories.NaturalTransformation.Properties.html#1611" class="Bound">F′</a> <a id="1679" href="Categories.NaturalTransformation.Properties.html#821" class="Bound">G</a>
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<a id="1683" href="Categories.NaturalTransformation.Properties.html#1597" class="Function">replaceˡ</a> <a id="1692" class="Symbol">{</a><a id="1693" href="Categories.NaturalTransformation.Properties.html#1693" class="Bound">F′</a><a id="1695" class="Symbol">}</a> <a id="1697" href="Categories.NaturalTransformation.Properties.html#1697" class="Bound">α</a> <a id="1699" href="Categories.NaturalTransformation.Properties.html#1699" class="Bound">F≃F′</a> <a id="1704" class="Symbol">=</a> <a id="1706" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="1715" class="Keyword">record</a>
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<a id="1726" class="Symbol">{</a> <a id="1728" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="1736" class="Symbol">=</a> <a id="1738" class="Symbol">λ</a> <a id="1740" href="Categories.NaturalTransformation.Properties.html#1740" class="Bound">X</a> <a id="1742" class="Symbol">→</a> <a id="1744" href="Categories.NaturalTransformation.Core.html#783" class="Function">η</a> <a id="1746" href="Categories.NaturalTransformation.Properties.html#1740" class="Bound">X</a> <a id="1748" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1750" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="1754" href="Categories.NaturalTransformation.Properties.html#1740" class="Bound">X</a>
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<a id="1760" class="Symbol">;</a> <a id="1762" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="1770" class="Symbol">=</a> <a id="1772" class="Symbol">λ</a> <a id="1774" class="Symbol">{</a><a id="1775" href="Categories.NaturalTransformation.Properties.html#1775" class="Bound">X</a> <a id="1777" href="Categories.NaturalTransformation.Properties.html#1777" class="Bound">Y</a><a id="1778" class="Symbol">}</a> <a id="1780" href="Categories.NaturalTransformation.Properties.html#1780" class="Bound">f</a> <a id="1782" class="Symbol">→</a> <a id="1784" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
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<a id="1796" class="Symbol">(</a><a id="1797" href="Categories.NaturalTransformation.Core.html#783" class="Function">η</a> <a id="1799" href="Categories.NaturalTransformation.Properties.html#1777" class="Bound">Y</a> <a id="1801" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1803" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="1807" href="Categories.NaturalTransformation.Properties.html#1777" class="Bound">Y</a><a id="1808" class="Symbol">)</a> <a id="1810" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1812" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1815" href="Categories.NaturalTransformation.Properties.html#1693" class="Bound">F′</a> <a id="1818" href="Categories.NaturalTransformation.Properties.html#1780" class="Bound">f</a> <a id="1820" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1823" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="1829" class="Symbol">(</a><a id="1830" href="Categories.NaturalTransformation.Core.html#827" class="Function">⇐.commute</a> <a id="1840" href="Categories.NaturalTransformation.Properties.html#1780" class="Bound">f</a><a id="1841" class="Symbol">)</a> <a id="1843" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="1851" href="Categories.NaturalTransformation.Core.html#783" class="Function">η</a> <a id="1853" href="Categories.NaturalTransformation.Properties.html#1777" class="Bound">Y</a> <a id="1855" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1857" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1860" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">F</a> <a id="1862" href="Categories.NaturalTransformation.Properties.html#1780" class="Bound">f</a> <a id="1864" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1866" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="1870" href="Categories.NaturalTransformation.Properties.html#1775" class="Bound">X</a> <a id="1875" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1878" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="1884" class="Symbol">(</a><a id="1885" href="Categories.NaturalTransformation.Core.html#827" class="Function">commute</a> <a id="1893" href="Categories.NaturalTransformation.Properties.html#1780" class="Bound">f</a><a id="1894" class="Symbol">)</a> <a id="1896" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1898" href="Categories.Category.Core.html#715" class="Function">assoc</a> <a id="1904" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
|
|||
|
|
<a id="1912" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1915" href="Categories.NaturalTransformation.Properties.html#821" class="Bound">G</a> <a id="1917" href="Categories.NaturalTransformation.Properties.html#1780" class="Bound">f</a> <a id="1919" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1921" href="Categories.NaturalTransformation.Core.html#783" class="Function">η</a> <a id="1923" href="Categories.NaturalTransformation.Properties.html#1775" class="Bound">X</a> <a id="1925" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1927" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="1931" href="Categories.NaturalTransformation.Properties.html#1775" class="Bound">X</a> <a id="1936" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
|
|||
|
|
<a id="1942" class="Symbol">}</a>
|
|||
|
|
<a id="1948" class="Keyword">where</a> <a id="1954" class="Keyword">open</a> <a id="1959" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Module">NaturalIsomorphism</a> <a id="1978" href="Categories.NaturalTransformation.Properties.html#1699" class="Bound">F≃F′</a>
|
|||
|
|
<a id="1993" class="Keyword">open</a> <a id="1998" href="Categories.NaturalTransformation.Core.html#466" class="Module">NaturalTransformation</a> <a id="2020" href="Categories.NaturalTransformation.Properties.html#1697" class="Bound">α</a>
|
|||
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|
|||
|
|
<a id="2025" href="Categories.NaturalTransformation.Properties.html#2025" class="Function">replaceʳ</a> <a id="2034" class="Symbol">:</a> <a id="2036" class="Symbol">∀</a> <a id="2038" class="Symbol">{</a><a id="2039" href="Categories.NaturalTransformation.Properties.html#2039" class="Bound">G′</a><a id="2041" class="Symbol">}</a> <a id="2043" class="Symbol">→</a> <a id="2045" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="2067" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">F</a> <a id="2069" href="Categories.NaturalTransformation.Properties.html#821" class="Bound">G</a> <a id="2071" class="Symbol">→</a> <a id="2073" href="Categories.NaturalTransformation.Properties.html#821" class="Bound">G</a> <a id="2075" href="Categories.NaturalTransformation.NaturalIsomorphism.html#3600" class="Function Operator">≃</a> <a id="2077" href="Categories.NaturalTransformation.Properties.html#2039" class="Bound">G′</a> <a id="2080" class="Symbol">→</a> <a id="2082" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="2104" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">F</a> <a id="2106" href="Categories.NaturalTransformation.Properties.html#2039" class="Bound">G′</a>
|
|||
|
|
<a id="2111" href="Categories.NaturalTransformation.Properties.html#2025" class="Function">replaceʳ</a> <a id="2120" class="Symbol">{</a><a id="2121" href="Categories.NaturalTransformation.Properties.html#2121" class="Bound">G′</a><a id="2123" class="Symbol">}</a> <a id="2125" href="Categories.NaturalTransformation.Properties.html#2125" class="Bound">α</a> <a id="2127" href="Categories.NaturalTransformation.Properties.html#2127" class="Bound">G≃G′</a> <a id="2132" class="Symbol">=</a> <a id="2134" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="2143" class="Keyword">record</a>
|
|||
|
|
<a id="2154" class="Symbol">{</a> <a id="2156" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="2164" class="Symbol">=</a> <a id="2166" class="Symbol">λ</a> <a id="2168" href="Categories.NaturalTransformation.Properties.html#2168" class="Bound">X</a> <a id="2170" class="Symbol">→</a> <a id="2172" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="2176" href="Categories.NaturalTransformation.Properties.html#2168" class="Bound">X</a> <a id="2178" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2180" href="Categories.NaturalTransformation.Core.html#783" class="Function">η</a> <a id="2182" href="Categories.NaturalTransformation.Properties.html#2168" class="Bound">X</a>
|
|||
|
|
<a id="2188" class="Symbol">;</a> <a id="2190" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="2198" class="Symbol">=</a> <a id="2200" class="Symbol">λ</a> <a id="2202" class="Symbol">{</a><a id="2203" href="Categories.NaturalTransformation.Properties.html#2203" class="Bound">X</a> <a id="2205" href="Categories.NaturalTransformation.Properties.html#2205" class="Bound">Y</a><a id="2206" class="Symbol">}</a> <a id="2208" href="Categories.NaturalTransformation.Properties.html#2208" class="Bound">f</a> <a id="2210" class="Symbol">→</a> <a id="2212" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
|
|||
|
|
<a id="2224" class="Symbol">(</a><a id="2225" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="2229" href="Categories.NaturalTransformation.Properties.html#2205" class="Bound">Y</a> <a id="2231" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2233" href="Categories.NaturalTransformation.Core.html#783" class="Function">η</a> <a id="2235" href="Categories.NaturalTransformation.Properties.html#2205" class="Bound">Y</a><a id="2236" class="Symbol">)</a> <a id="2238" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2240" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="2243" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">F</a> <a id="2245" href="Categories.NaturalTransformation.Properties.html#2208" class="Bound">f</a> <a id="2247" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2250" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="2256" class="Symbol">(</a><a id="2257" href="Categories.NaturalTransformation.Core.html#827" class="Function">commute</a> <a id="2265" href="Categories.NaturalTransformation.Properties.html#2208" class="Bound">f</a><a id="2266" class="Symbol">)</a> <a id="2268" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
|
|||
|
|
<a id="2276" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="2280" href="Categories.NaturalTransformation.Properties.html#2205" class="Bound">Y</a> <a id="2282" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2284" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="2287" href="Categories.NaturalTransformation.Properties.html#821" class="Bound">G</a> <a id="2289" href="Categories.NaturalTransformation.Properties.html#2208" class="Bound">f</a> <a id="2291" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2293" href="Categories.NaturalTransformation.Core.html#783" class="Function">η</a> <a id="2295" href="Categories.NaturalTransformation.Properties.html#2203" class="Bound">X</a> <a id="2299" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2302" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="2308" class="Symbol">(</a><a id="2309" href="Categories.NaturalTransformation.Core.html#827" class="Function">⇒.commute</a> <a id="2319" href="Categories.NaturalTransformation.Properties.html#2208" class="Bound">f</a><a id="2320" class="Symbol">)</a> <a id="2322" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2324" href="Categories.Category.Core.html#715" class="Function">assoc</a> <a id="2330" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
|
|||
|
|
<a id="2338" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="2341" href="Categories.NaturalTransformation.Properties.html#2121" class="Bound">G′</a> <a id="2344" href="Categories.NaturalTransformation.Properties.html#2208" class="Bound">f</a> <a id="2346" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2348" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="2352" href="Categories.NaturalTransformation.Properties.html#2203" class="Bound">X</a> <a id="2354" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2356" href="Categories.NaturalTransformation.Core.html#783" class="Function">η</a> <a id="2358" href="Categories.NaturalTransformation.Properties.html#2203" class="Bound">X</a> <a id="2361" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
|
|||
|
|
<a id="2367" class="Symbol">}</a>
|
|||
|
|
<a id="2373" class="Keyword">where</a> <a id="2379" class="Keyword">open</a> <a id="2384" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Module">NaturalIsomorphism</a> <a id="2403" href="Categories.NaturalTransformation.Properties.html#2127" class="Bound">G≃G′</a>
|
|||
|
|
<a id="2418" class="Keyword">open</a> <a id="2423" href="Categories.NaturalTransformation.Core.html#466" class="Module">NaturalTransformation</a> <a id="2445" href="Categories.NaturalTransformation.Properties.html#2125" class="Bound">α</a>
|
|||
|
|
|
|||
|
|
<a id="2450" class="Keyword">module</a> <a id="2457" href="Categories.NaturalTransformation.Properties.html#2457" class="Module">_</a> <a id="2459" class="Symbol">(</a><a id="2460" href="Categories.NaturalTransformation.Properties.html#2460" class="Bound">F</a> <a id="2462" class="Symbol">:</a> <a id="2464" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="2474" href="Categories.NaturalTransformation.Properties.html#785" class="Generalizable">C</a> <a id="2476" href="Categories.NaturalTransformation.Properties.html#787" class="Generalizable">D</a> <a id="2478" href="Categories.NaturalTransformation.Properties.html#789" class="Generalizable">E</a><a id="2479" class="Symbol">)</a> <a id="2481" class="Keyword">where</a>
|
|||
|
|
|
|||
|
|
<a id="2490" class="Comment">-- there is natural transformation between two partially applied bifunctors.</a>
|
|||
|
|
|
|||
|
|
<a id="2570" href="Categories.NaturalTransformation.Properties.html#2570" class="Function">appˡ-nat</a> <a id="2579" class="Symbol">:</a> <a id="2581" class="Symbol">∀</a> <a id="2583" class="Symbol">{</a><a id="2584" href="Categories.NaturalTransformation.Properties.html#2584" class="Bound">X</a> <a id="2586" href="Categories.NaturalTransformation.Properties.html#2586" class="Bound">Y</a><a id="2587" class="Symbol">}</a> <a id="2589" class="Symbol">→</a> <a id="2591" class="Symbol">(</a><a id="2592" href="Categories.NaturalTransformation.Properties.html#2592" class="Bound">f</a> <a id="2594" class="Symbol">:</a> <a id="2596" href="Categories.Category.Core.html#575" class="Field Operator">Category._⇒_</a> <a id="2609" href="Categories.NaturalTransformation.Properties.html#2474" class="Bound">C</a> <a id="2611" href="Categories.NaturalTransformation.Properties.html#2584" class="Bound">X</a> <a id="2613" href="Categories.NaturalTransformation.Properties.html#2586" class="Bound">Y</a><a id="2614" class="Symbol">)</a> <a id="2616" class="Symbol">→</a> <a id="2618" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="2640" class="Symbol">(</a><a id="2641" href="Categories.Functor.Bifunctor.html#874" class="Function">appˡ</a> <a id="2646" href="Categories.NaturalTransformation.Properties.html#2460" class="Bound">F</a> <a id="2648" href="Categories.NaturalTransformation.Properties.html#2584" class="Bound">X</a><a id="2649" class="Symbol">)</a> <a id="2651" class="Symbol">(</a><a id="2652" href="Categories.Functor.Bifunctor.html#874" class="Function">appˡ</a> <a id="2657" href="Categories.NaturalTransformation.Properties.html#2460" class="Bound">F</a> <a id="2659" href="Categories.NaturalTransformation.Properties.html#2586" class="Bound">Y</a><a id="2660" class="Symbol">)</a>
|
|||
|
|
<a id="2664" href="Categories.NaturalTransformation.Properties.html#2570" class="Function">appˡ-nat</a> <a id="2673" href="Categories.NaturalTransformation.Properties.html#2673" class="Bound">f</a> <a id="2675" class="Symbol">=</a> <a id="2677" href="Categories.NaturalTransformation.Properties.html#2460" class="Bound">F</a> <a id="2679" href="Categories.NaturalTransformation.Core.html#3439" class="Function Operator">∘ˡ</a> <a id="2682" class="Symbol">(</a><a id="2683" href="Categories.Functor.Construction.Constant.html#964" class="Function">constNat</a> <a id="2692" href="Categories.NaturalTransformation.Properties.html#2673" class="Bound">f</a> <a id="2694" href="Categories.Category.Product.html#2966" class="Function Operator">※ⁿ</a> <a id="2697" href="Categories.NaturalTransformation.Properties.html#559" class="Function">idN</a><a id="2700" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="2705" href="Categories.NaturalTransformation.Properties.html#2705" class="Function">appʳ-nat</a> <a id="2714" class="Symbol">:</a> <a id="2716" class="Symbol">∀</a> <a id="2718" class="Symbol">{</a><a id="2719" href="Categories.NaturalTransformation.Properties.html#2719" class="Bound">X</a> <a id="2721" href="Categories.NaturalTransformation.Properties.html#2721" class="Bound">Y</a><a id="2722" class="Symbol">}</a> <a id="2724" class="Symbol">→</a> <a id="2726" class="Symbol">(</a><a id="2727" href="Categories.NaturalTransformation.Properties.html#2727" class="Bound">f</a> <a id="2729" class="Symbol">:</a> <a id="2731" href="Categories.Category.Core.html#575" class="Field Operator">Category._⇒_</a> <a id="2744" href="Categories.NaturalTransformation.Properties.html#2476" class="Bound">D</a> <a id="2746" href="Categories.NaturalTransformation.Properties.html#2719" class="Bound">X</a> <a id="2748" href="Categories.NaturalTransformation.Properties.html#2721" class="Bound">Y</a><a id="2749" class="Symbol">)</a> <a id="2751" class="Symbol">→</a> <a id="2753" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="2775" class="Symbol">(</a><a id="2776" href="Categories.Functor.Bifunctor.html#938" class="Function">appʳ</a> <a id="2781" href="Categories.NaturalTransformation.Properties.html#2460" class="Bound">F</a> <a id="2783" href="Categories.NaturalTransformation.Properties.html#2719" class="Bound">X</a><a id="2784" class="Symbol">)</a> <a id="2786" class="Symbol">(</a><a id="2787" href="Categories.Functor.Bifunctor.html#938" class="Function">appʳ</a> <a id="2792" href="Categories.NaturalTransformation.Properties.html#2460" class="Bound">F</a> <a id="2794" href="Categories.NaturalTransformation.Properties.html#2721" class="Bound">Y</a><a id="2795" class="Symbol">)</a>
|
|||
|
|
<a id="2799" href="Categories.NaturalTransformation.Properties.html#2705" class="Function">appʳ-nat</a> <a id="2808" href="Categories.NaturalTransformation.Properties.html#2808" class="Bound">f</a> <a id="2810" class="Symbol">=</a> <a id="2812" href="Categories.NaturalTransformation.Properties.html#2460" class="Bound">F</a> <a id="2814" href="Categories.NaturalTransformation.Core.html#3439" class="Function Operator">∘ˡ</a> <a id="2817" class="Symbol">(</a><a id="2818" href="Categories.NaturalTransformation.Properties.html#559" class="Function">idN</a> <a id="2822" href="Categories.Category.Product.html#2966" class="Function Operator">※ⁿ</a> <a id="2825" href="Categories.Functor.Construction.Constant.html#964" class="Function">constNat</a> <a id="2834" href="Categories.NaturalTransformation.Properties.html#2808" class="Bound">f</a><a id="2835" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="2838" class="Keyword">module</a> <a id="2845" href="Categories.NaturalTransformation.Properties.html#2845" class="Module">_</a> <a id="2847" class="Symbol">{</a><a id="2848" href="Categories.NaturalTransformation.Properties.html#2848" class="Bound">F</a> <a id="2850" href="Categories.NaturalTransformation.Properties.html#2850" class="Bound">G</a> <a id="2852" class="Symbol">:</a> <a id="2854" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="2864" href="Categories.NaturalTransformation.Properties.html#785" class="Generalizable">C</a> <a id="2866" href="Categories.NaturalTransformation.Properties.html#787" class="Generalizable">D</a> <a id="2868" href="Categories.NaturalTransformation.Properties.html#789" class="Generalizable">E</a><a id="2869" class="Symbol">}</a> <a id="2871" class="Symbol">(</a><a id="2872" href="Categories.NaturalTransformation.Properties.html#2872" class="Bound">α</a> <a id="2874" class="Symbol">:</a> <a id="2876" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="2898" href="Categories.NaturalTransformation.Properties.html#2848" class="Bound">F</a> <a id="2900" href="Categories.NaturalTransformation.Properties.html#2850" class="Bound">G</a><a id="2901" class="Symbol">)</a> <a id="2903" class="Keyword">where</a>
|
|||
|
|
<a id="2911" class="Keyword">private</a>
|
|||
|
|
<a id="2923" class="Keyword">module</a> <a id="2930" href="Categories.NaturalTransformation.Properties.html#2930" class="Module">C</a> <a id="2932" class="Symbol">=</a> <a id="2934" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="2943" href="Categories.NaturalTransformation.Properties.html#2864" class="Bound">C</a>
|
|||
|
|
<a id="2949" class="Keyword">module</a> <a id="2956" href="Categories.NaturalTransformation.Properties.html#2956" class="Module">D</a> <a id="2958" class="Symbol">=</a> <a id="2960" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="2969" href="Categories.NaturalTransformation.Properties.html#2866" class="Bound">D</a>
|
|||
|
|
<a id="2975" class="Keyword">module</a> <a id="2982" href="Categories.NaturalTransformation.Properties.html#2982" class="Module">E</a> <a id="2984" class="Symbol">=</a> <a id="2986" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="2995" href="Categories.NaturalTransformation.Properties.html#2868" class="Bound">E</a>
|
|||
|
|
|
|||
|
|
<a id="3000" href="Categories.NaturalTransformation.Properties.html#3000" class="Function">appˡ′</a> <a id="3006" class="Symbol">:</a> <a id="3008" class="Symbol">∀</a> <a id="3010" href="Categories.NaturalTransformation.Properties.html#3010" class="Bound">X</a> <a id="3012" class="Symbol">→</a> <a id="3014" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="3036" class="Symbol">(</a><a id="3037" href="Categories.Functor.Bifunctor.html#874" class="Function">appˡ</a> <a id="3042" href="Categories.NaturalTransformation.Properties.html#2848" class="Bound">F</a> <a id="3044" href="Categories.NaturalTransformation.Properties.html#3010" class="Bound">X</a><a id="3045" class="Symbol">)</a> <a id="3047" class="Symbol">(</a><a id="3048" href="Categories.Functor.Bifunctor.html#874" class="Function">appˡ</a> <a id="3053" href="Categories.NaturalTransformation.Properties.html#2850" class="Bound">G</a> <a id="3055" href="Categories.NaturalTransformation.Properties.html#3010" class="Bound">X</a><a id="3056" class="Symbol">)</a>
|
|||
|
|
<a id="3060" href="Categories.NaturalTransformation.Properties.html#3000" class="Function">appˡ′</a> <a id="3066" href="Categories.NaturalTransformation.Properties.html#3066" class="Bound">X</a> <a id="3068" class="Symbol">=</a> <a id="3070" href="Categories.NaturalTransformation.Properties.html#2872" class="Bound">α</a> <a id="3072" href="Categories.NaturalTransformation.Core.html#2919" class="Function Operator">∘ₕ</a> <a id="3075" href="Categories.NaturalTransformation.Properties.html#559" class="Function">idN</a>
|
|||
|
|
|
|||
|
|
<a id="3082" href="Categories.NaturalTransformation.Properties.html#3082" class="Function">appʳ′</a> <a id="3088" class="Symbol">:</a> <a id="3090" class="Symbol">∀</a> <a id="3092" href="Categories.NaturalTransformation.Properties.html#3092" class="Bound">X</a> <a id="3094" class="Symbol">→</a> <a id="3096" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="3118" class="Symbol">(</a><a id="3119" href="Categories.Functor.Bifunctor.html#938" class="Function">appʳ</a> <a id="3124" href="Categories.NaturalTransformation.Properties.html#2848" class="Bound">F</a> <a id="3126" href="Categories.NaturalTransformation.Properties.html#3092" class="Bound">X</a><a id="3127" class="Symbol">)</a> <a id="3129" class="Symbol">(</a><a id="3130" href="Categories.Functor.Bifunctor.html#938" class="Function">appʳ</a> <a id="3135" href="Categories.NaturalTransformation.Properties.html#2850" class="Bound">G</a> <a id="3137" href="Categories.NaturalTransformation.Properties.html#3092" class="Bound">X</a><a id="3138" class="Symbol">)</a>
|
|||
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<a id="3142" href="Categories.NaturalTransformation.Properties.html#3082" class="Function">appʳ′</a> <a id="3148" href="Categories.NaturalTransformation.Properties.html#3148" class="Bound">X</a> <a id="3150" class="Symbol">=</a> <a id="3152" href="Categories.NaturalTransformation.Properties.html#2872" class="Bound">α</a> <a id="3154" href="Categories.NaturalTransformation.Core.html#2919" class="Function Operator">∘ₕ</a> <a id="3157" href="Categories.NaturalTransformation.Properties.html#559" class="Function">idN</a>
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<a id="3162" class="Comment">-- unlift universe level</a>
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<a id="3187" class="Keyword">module</a> <a id="3194" href="Categories.NaturalTransformation.Properties.html#3194" class="Module">_</a> <a id="3196" class="Symbol">{</a><a id="3197" href="Categories.NaturalTransformation.Properties.html#3197" class="Bound">c</a> <a id="3199" href="Categories.NaturalTransformation.Properties.html#3199" class="Bound">ℓ</a> <a id="3201" href="Categories.NaturalTransformation.Properties.html#3201" class="Bound">ℓ′</a> <a id="3204" href="Categories.NaturalTransformation.Properties.html#3204" class="Bound">e</a><a id="3205" class="Symbol">}</a> <a id="3207" class="Symbol">{</a><a id="3208" href="Categories.NaturalTransformation.Properties.html#3208" class="Bound">F</a> <a id="3210" href="Categories.NaturalTransformation.Properties.html#3210" class="Bound">G</a> <a id="3212" class="Symbol">:</a> <a id="3214" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="3222" href="Categories.NaturalTransformation.Properties.html#785" class="Generalizable">C</a> <a id="3224" class="Symbol">(</a><a id="3225" href="Categories.Category.Instance.Setoids.html#555" class="Function">Setoids</a> <a id="3233" href="Categories.NaturalTransformation.Properties.html#3197" class="Bound">c</a> <a id="3235" href="Categories.NaturalTransformation.Properties.html#3199" class="Bound">ℓ</a><a id="3236" class="Symbol">)}</a> <a id="3239" class="Symbol">(</a><a id="3240" href="Categories.NaturalTransformation.Properties.html#3240" class="Bound">α</a> <a id="3242" class="Symbol">:</a> <a id="3244" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="3266" class="Symbol">(</a><a id="3267" href="Categories.Functor.Construction.LiftSetoids.html#1046" class="Function">LiftSetoids</a> <a id="3279" href="Categories.NaturalTransformation.Properties.html#3201" class="Bound">ℓ′</a> <a id="3282" href="Categories.NaturalTransformation.Properties.html#3204" class="Bound">e</a> <a id="3284" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="3287" href="Categories.NaturalTransformation.Properties.html#3208" class="Bound">F</a><a id="3288" class="Symbol">)</a> <a id="3290" class="Symbol">(</a><a id="3291" href="Categories.Functor.Construction.LiftSetoids.html#1046" class="Function">LiftSetoids</a> <a id="3303" href="Categories.NaturalTransformation.Properties.html#3201" class="Bound">ℓ′</a> <a id="3306" href="Categories.NaturalTransformation.Properties.html#3204" class="Bound">e</a> <a id="3308" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="3311" href="Categories.NaturalTransformation.Properties.html#3210" class="Bound">G</a><a id="3312" class="Symbol">))</a> <a id="3315" class="Keyword">where</a>
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<a id="3323" class="Keyword">open</a> <a id="3328" href="Categories.NaturalTransformation.Core.html#466" class="Module">NaturalTransformation</a> <a id="3350" href="Categories.NaturalTransformation.Properties.html#3240" class="Bound">α</a>
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<a id="3354" class="Keyword">open</a> <a id="3359" href="Function.Bundles.html#2043" class="Module">Func</a>
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|||
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|
|||
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<a id="3367" href="Categories.NaturalTransformation.Properties.html#3367" class="Function">unlift-nat</a> <a id="3378" class="Symbol">:</a> <a id="3380" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="3402" href="Categories.NaturalTransformation.Properties.html#3208" class="Bound">F</a> <a id="3404" href="Categories.NaturalTransformation.Properties.html#3210" class="Bound">G</a>
|
|||
|
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<a id="3408" href="Categories.NaturalTransformation.Properties.html#3367" class="Function">unlift-nat</a> <a id="3419" class="Symbol">=</a> <a id="3421" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="3430" class="Keyword">record</a>
|
|||
|
|
<a id="3441" class="Symbol">{</a> <a id="3443" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="3451" class="Symbol">=</a> <a id="3453" class="Symbol">λ</a> <a id="3455" href="Categories.NaturalTransformation.Properties.html#3455" class="Bound">X</a> <a id="3457" class="Symbol">→</a> <a id="3459" class="Keyword">record</a>
|
|||
|
|
<a id="3472" class="Symbol">{</a> <a id="3474" href="Function.Bundles.html#2094" class="Field">to</a> <a id="3477" class="Symbol">=</a> <a id="3479" class="Symbol">λ</a> <a id="3481" href="Categories.NaturalTransformation.Properties.html#3481" class="Bound">x</a> <a id="3483" class="Symbol">→</a> <a id="3485" href="Level.html#479" class="Field">lower</a> <a id="3491" class="Symbol">(</a><a id="3492" href="Function.Bundles.html#2094" class="Field">to</a> <a id="3495" class="Symbol">(</a><a id="3496" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="3498" href="Categories.NaturalTransformation.Properties.html#3455" class="Bound">X</a><a id="3499" class="Symbol">)</a> <a id="3501" class="Symbol">(</a><a id="3502" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="3507" href="Categories.NaturalTransformation.Properties.html#3481" class="Bound">x</a><a id="3508" class="Symbol">))</a>
|
|||
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<a id="3517" class="Symbol">;</a> <a id="3519" href="Function.Bundles.html#2113" class="Field">cong</a> <a id="3524" class="Symbol">=</a> <a id="3526" class="Symbol">λ</a> <a id="3528" href="Categories.NaturalTransformation.Properties.html#3528" class="Bound">eq</a> <a id="3531" class="Symbol">→</a> <a id="3533" href="Level.html#479" class="Field">lower</a> <a id="3539" class="Symbol">(</a><a id="3540" href="Function.Bundles.html#2113" class="Field">cong</a> <a id="3545" class="Symbol">(</a><a id="3546" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="3548" href="Categories.NaturalTransformation.Properties.html#3455" class="Bound">X</a><a id="3549" class="Symbol">)</a> <a id="3551" class="Symbol">(</a><a id="3552" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="3557" href="Categories.NaturalTransformation.Properties.html#3528" class="Bound">eq</a><a id="3559" class="Symbol">))</a>
|
|||
|
|
<a id="3568" class="Symbol">}</a>
|
|||
|
|
<a id="3574" class="Symbol">;</a> <a id="3576" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="3584" class="Symbol">=</a> <a id="3586" class="Symbol">λ</a> <a id="3588" href="Categories.NaturalTransformation.Properties.html#3588" class="Bound">f</a> <a id="3590" class="Symbol">→</a> <a id="3592" href="Level.html#479" class="Field">lower</a> <a id="3598" class="Symbol">(</a><a id="3599" href="Categories.NaturalTransformation.Core.html#827" class="Field">commute</a> <a id="3607" href="Categories.NaturalTransformation.Properties.html#3588" class="Bound">f</a><a id="3608" class="Symbol">)</a>
|
|||
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|
<a id="3614" class="Symbol">}</a>
|
|||
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</pre></body></html>
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