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<html><head><meta charset="utf-8"><title>Categories.Adjoint.Equivalents</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="36" class="Keyword">module</a> <a id="43" href="Categories.Adjoint.Equivalents.html" class="Module">Categories.Adjoint.Equivalents</a> <a id="74" class="Keyword">where</a>
<a id="81" class="Comment">-- Theorems about equivalent formulations to Adjoint</a>
<a id="134" class="Comment">-- (though some have caveats)</a>
<a id="165" class="Keyword">open</a> <a id="170" class="Keyword">import</a> <a id="177" href="Level.html" class="Module">Level</a>
<a id="184" class="Keyword">open</a> <a id="189" class="Keyword">import</a> <a id="196" href="Data.Product.html" class="Module">Data.Product</a> <a id="209" class="Keyword">using</a> <a id="215" class="Symbol">(</a><a id="216" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="219" class="Symbol">;</a> <a id="221" href="Data.Product.Base.html#1618" class="Function Operator">_×_</a><a id="224" class="Symbol">)</a>
<a id="226" class="Keyword">open</a> <a id="231" class="Keyword">import</a> <a id="238" href="Function.html" class="Module">Function</a> <a id="247" class="Keyword">using</a> <a id="253" class="Symbol">(</a><a id="254" href="Function.Base.html#1974" class="Function Operator">_$_</a><a id="257" class="Symbol">)</a> <a id="259" class="Keyword">renaming</a> <a id="268" class="Symbol">(</a><a id="269" href="Function.Base.html#1115" class="Function Operator">_∘_</a> <a id="273" class="Symbol">to</a> <a id="276" class="Function Operator">_∙_</a><a id="279" class="Symbol">)</a>
<a id="281" class="Keyword">open</a> <a id="286" class="Keyword">import</a> <a id="293" href="Function.Bundles.html" class="Module">Function.Bundles</a> <a id="310" class="Keyword">using</a> <a id="316" class="Symbol">(</a><a id="317" href="Function.Bundles.html#4752" class="Record">Equivalence</a><a id="328" class="Symbol">;</a> <a id="330" href="Function.Bundles.html#5375" class="Record">LeftInverse</a><a id="341" class="Symbol">;</a> <a id="343" href="Function.Bundles.html#2043" class="Record">Func</a><a id="347" class="Symbol">;</a> <a id="349" href="Function.Bundles.html#15133" class="Function Operator">_⟨$⟩_</a><a id="354" class="Symbol">)</a>
<a id="356" class="Keyword">open</a> <a id="361" class="Keyword">import</a> <a id="368" href="Relation.Binary.html" class="Module">Relation.Binary</a> <a id="384" class="Keyword">using</a> <a id="390" class="Symbol">(</a><a id="391" href="Relation.Binary.Core.html#896" class="Function">Rel</a><a id="394" class="Symbol">;</a> <a id="396" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a><a id="409" class="Symbol">;</a> <a id="411" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a><a id="417" class="Symbol">)</a>
<a id="420" class="Comment">-- be explicit in imports to &#39;see&#39; where the information comes from</a>
<a id="488" class="Keyword">open</a> <a id="493" class="Keyword">import</a> <a id="500" href="Categories.Adjoint.html" class="Module">Categories.Adjoint</a> <a id="519" class="Keyword">using</a> <a id="525" class="Symbol">(</a><a id="526" href="Categories.Adjoint.html#1260" class="Record">Adjoint</a><a id="533" class="Symbol">;</a> <a id="535" href="Categories.Adjoint.html#7818" class="Function Operator">_⊣_</a><a id="538" class="Symbol">)</a>
<a id="540" class="Keyword">open</a> <a id="545" class="Keyword">import</a> <a id="552" href="Categories.Category.Core.html" class="Module">Categories.Category.Core</a> <a id="577" class="Keyword">using</a> <a id="583" class="Symbol">(</a><a id="584" href="Categories.Category.Core.html#442" class="Record">Category</a><a id="592" class="Symbol">)</a>
<a id="594" class="Keyword">open</a> <a id="599" class="Keyword">import</a> <a id="606" href="Categories.Category.Product.html" class="Module">Categories.Category.Product</a> <a id="634" class="Keyword">using</a> <a id="640" class="Symbol">(</a><a id="641" href="Categories.Category.Product.html#745" class="Function">Product</a><a id="648" class="Symbol">;</a> <a id="650" href="Categories.Category.Product.html#1962" class="Function Operator">_⁂_</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="Categories.Category.Instance.Setoids.html" class="Module">Categories.Category.Instance.Setoids</a> <a id="704" class="Keyword">using</a> <a id="710" class="Symbol">(</a><a id="711" href="Categories.Category.Instance.Setoids.html#555" class="Function">Setoids</a><a id="718" class="Symbol">)</a>
<a id="720" class="Keyword">open</a> <a id="725" class="Keyword">import</a> <a id="732" href="Categories.Morphism.html" class="Module">Categories.Morphism</a> <a id="752" class="Keyword">using</a> <a id="758" class="Symbol">(</a><a id="759" href="Categories.Morphism.html#1528" class="Record">Iso</a><a id="762" class="Symbol">)</a>
<a id="764" class="Keyword">open</a> <a id="769" class="Keyword">import</a> <a id="776" href="Categories.Functor.html" class="Module">Categories.Functor</a> <a id="795" class="Keyword">using</a> <a id="801" class="Symbol">(</a><a id="802" href="Categories.Functor.Core.html#248" class="Record">Functor</a><a id="809" class="Symbol">;</a> <a id="811" href="Categories.Functor.html#747" class="Function Operator">_∘F_</a><a id="815" class="Symbol">)</a> <a id="817" class="Keyword">renaming</a> <a id="826" class="Symbol">(</a><a id="827" href="Categories.Functor.html#349" class="Function">id</a> <a id="830" class="Symbol">to</a> <a id="833" class="Function">idF</a><a id="836" class="Symbol">)</a>
<a id="838" class="Keyword">open</a> <a id="843" class="Keyword">import</a> <a id="850" href="Categories.Functor.Bifunctor.html" class="Module">Categories.Functor.Bifunctor</a> <a id="879" class="Keyword">using</a> <a id="885" class="Symbol">(</a><a id="886" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a><a id="895" class="Symbol">)</a>
<a id="897" class="Keyword">open</a> <a id="902" class="Keyword">import</a> <a id="909" href="Categories.Functor.Hom.html" class="Module">Categories.Functor.Hom</a> <a id="932" class="Keyword">using</a> <a id="938" class="Symbol">(</a><a id="939" href="Categories.Functor.Hom.html#1578" class="Function Operator">Hom[_][-,-]</a><a id="950" class="Symbol">)</a>
<a id="952" class="Keyword">open</a> <a id="957" class="Keyword">import</a> <a id="964" href="Categories.Functor.Construction.LiftSetoids.html" class="Module">Categories.Functor.Construction.LiftSetoids</a>
<a id="1008" class="Keyword">open</a> <a id="1013" class="Keyword">import</a> <a id="1020" href="Categories.NaturalTransformation.html" class="Module">Categories.NaturalTransformation</a> <a id="1053" class="Keyword">using</a> <a id="1059" class="Symbol">(</a><a id="1060" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a><a id="1081" class="Symbol">;</a> <a id="1083" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a><a id="1091" class="Symbol">;</a> <a id="1093" href="Categories.NaturalTransformation.Core.html#2919" class="Function Operator">_∘ₕ_</a><a id="1097" class="Symbol">;</a> <a id="1099" href="Categories.NaturalTransformation.Core.html#2439" class="Function Operator">_∘ᵥ_</a><a id="1103" class="Symbol">;</a> <a id="1105" href="Categories.NaturalTransformation.Core.html#3439" class="Function Operator">_∘ˡ_</a><a id="1109" class="Symbol">;</a> <a id="1111" href="Categories.NaturalTransformation.Core.html#3784" class="Function Operator">_∘ʳ_</a><a id="1115" class="Symbol">)</a>
<a id="1119" class="Keyword">renaming</a> <a id="1128" class="Symbol">(</a><a id="1129" href="Categories.NaturalTransformation.Core.html#2132" class="Function">id</a> <a id="1132" class="Symbol">to</a> <a id="1135" class="Function">idN</a><a id="1138" class="Symbol">)</a>
<a id="1140" class="Keyword">open</a> <a id="1145" class="Keyword">import</a> <a id="1152" href="Categories.NaturalTransformation.NaturalIsomorphism.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism</a>
<a id="1206" class="Keyword">using</a> <a id="1212" class="Symbol">(</a><a id="1213" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a><a id="1231" class="Symbol">;</a> <a id="1233" href="Categories.NaturalTransformation.NaturalIsomorphism.html#6216" class="Function">unitorˡ</a><a id="1240" class="Symbol">;</a> <a id="1242" href="Categories.NaturalTransformation.NaturalIsomorphism.html#6312" class="Function">unitorʳ</a><a id="1249" class="Symbol">;</a> <a id="1251" href="Categories.NaturalTransformation.NaturalIsomorphism.html#7073" class="Function">associator</a><a id="1261" class="Symbol">;</a> <a id="1263" href="Categories.NaturalTransformation.NaturalIsomorphism.html#3600" class="Function Operator">_≃_</a><a id="1266" class="Symbol">)</a>
<a id="1268" class="Keyword">import</a> <a id="1275" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="1305" class="Symbol">as</a> <a id="1308" class="Module">MR</a>
<a id="1312" class="Keyword">private</a>
<a id="1322" class="Keyword">variable</a>
<a id="1335" href="Categories.Adjoint.Equivalents.html#1335" class="Generalizable">o</a> <a id="1337" href="Categories.Adjoint.Equivalents.html#1337" class="Generalizable">o′</a> <a id="1340" href="Categories.Adjoint.Equivalents.html#1340" class="Generalizable">o″</a> <a id="1343" href="Categories.Adjoint.Equivalents.html#1343" class="Generalizable">ℓ</a> <a id="1345" href="Categories.Adjoint.Equivalents.html#1345" class="Generalizable">ℓ′</a> <a id="1348" href="Categories.Adjoint.Equivalents.html#1348" class="Generalizable">ℓ″</a> <a id="1351" href="Categories.Adjoint.Equivalents.html#1351" class="Generalizable">e</a> <a id="1353" href="Categories.Adjoint.Equivalents.html#1353" class="Generalizable">e′</a> <a id="1356" href="Categories.Adjoint.Equivalents.html#1356" class="Generalizable">e″</a> <a id="1359" class="Symbol">:</a> <a id="1361" href="Agda.Primitive.html#742" class="Postulate">Level</a>
<a id="1371" href="Categories.Adjoint.Equivalents.html#1371" class="Generalizable">C</a> <a id="1373" href="Categories.Adjoint.Equivalents.html#1373" class="Generalizable">D</a> <a id="1375" href="Categories.Adjoint.Equivalents.html#1375" class="Generalizable">E</a> <a id="1377" class="Symbol">:</a> <a id="1379" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="1388" href="Categories.Adjoint.Equivalents.html#1335" class="Generalizable">o</a> <a id="1390" href="Categories.Adjoint.Equivalents.html#1343" class="Generalizable">ℓ</a> <a id="1392" href="Categories.Adjoint.Equivalents.html#1351" class="Generalizable">e</a>
<a id="1395" class="Comment">-- a special case of the natural isomorphism in which homsets in C and D have the same</a>
<a id="1482" class="Comment">-- universe level. therefore there is no need to lift Setoids to the same level.</a>
<a id="1563" class="Comment">-- this is helpful when combining with Yoneda lemma.</a>
<a id="1616" class="Keyword">module</a> <a id="1623" href="Categories.Adjoint.Equivalents.html#1623" class="Module">_</a> <a id="1625" class="Symbol">{</a><a id="1626" href="Categories.Adjoint.Equivalents.html#1626" class="Bound">C</a> <a id="1628" class="Symbol">:</a> <a id="1630" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="1639" href="Categories.Adjoint.Equivalents.html#1335" class="Generalizable">o</a> <a id="1641" href="Categories.Adjoint.Equivalents.html#1343" class="Generalizable">ℓ</a> <a id="1643" href="Categories.Adjoint.Equivalents.html#1351" class="Generalizable">e</a><a id="1644" class="Symbol">}</a> <a id="1646" class="Symbol">{</a><a id="1647" href="Categories.Adjoint.Equivalents.html#1647" class="Bound">D</a> <a id="1649" class="Symbol">:</a> <a id="1651" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="1660" href="Categories.Adjoint.Equivalents.html#1337" class="Generalizable">o′</a> <a id="1663" href="Categories.Adjoint.Equivalents.html#1343" class="Generalizable">ℓ</a> <a id="1665" href="Categories.Adjoint.Equivalents.html#1351" class="Generalizable">e</a><a id="1666" class="Symbol">}</a> <a id="1668" class="Symbol">{</a><a id="1669" href="Categories.Adjoint.Equivalents.html#1669" class="Bound">L</a> <a id="1671" class="Symbol">:</a> <a id="1673" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="1681" href="Categories.Adjoint.Equivalents.html#1626" class="Bound">C</a> <a id="1683" href="Categories.Adjoint.Equivalents.html#1647" class="Bound">D</a><a id="1684" class="Symbol">}</a> <a id="1686" class="Symbol">{</a><a id="1687" href="Categories.Adjoint.Equivalents.html#1687" class="Bound">R</a> <a id="1689" class="Symbol">:</a> <a id="1691" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="1699" href="Categories.Adjoint.Equivalents.html#1647" class="Bound">D</a> <a id="1701" href="Categories.Adjoint.Equivalents.html#1626" class="Bound">C</a><a id="1702" class="Symbol">}</a> <a id="1704" class="Keyword">where</a>
<a id="1712" class="Keyword">private</a>
<a id="1724" class="Keyword">module</a> <a id="1731" href="Categories.Adjoint.Equivalents.html#1731" class="Module">C</a> <a id="1733" class="Symbol">=</a> <a id="1735" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1744" href="Categories.Adjoint.Equivalents.html#1626" class="Bound">C</a>
<a id="1750" class="Keyword">module</a> <a id="1757" href="Categories.Adjoint.Equivalents.html#1757" class="Module">D</a> <a id="1759" class="Symbol">=</a> <a id="1761" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1770" href="Categories.Adjoint.Equivalents.html#1647" class="Bound">D</a>
<a id="1776" class="Keyword">module</a> <a id="1783" href="Categories.Adjoint.Equivalents.html#1783" class="Module">L</a> <a id="1785" class="Symbol">=</a> <a id="1787" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1795" href="Categories.Adjoint.Equivalents.html#1669" class="Bound">L</a>
<a id="1801" class="Keyword">module</a> <a id="1808" href="Categories.Adjoint.Equivalents.html#1808" class="Module">R</a> <a id="1810" class="Symbol">=</a> <a id="1812" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1820" href="Categories.Adjoint.Equivalents.html#1687" class="Bound">R</a>
<a id="1825" class="Keyword">module</a> <a id="1832" href="Categories.Adjoint.Equivalents.html#1832" class="Module">_</a> <a id="1834" class="Symbol">(</a><a id="1835" href="Categories.Adjoint.Equivalents.html#1835" class="Bound">adjoint</a> <a id="1843" class="Symbol">:</a> <a id="1845" href="Categories.Adjoint.Equivalents.html#1669" class="Bound">L</a> <a id="1847" href="Categories.Adjoint.html#7818" class="Function Operator">⊣</a> <a id="1849" href="Categories.Adjoint.Equivalents.html#1687" class="Bound">R</a><a id="1850" class="Symbol">)</a> <a id="1852" class="Keyword">where</a>
<a id="1862" class="Keyword">open</a> <a id="1867" href="Categories.Adjoint.html#1260" class="Module">Adjoint</a> <a id="1875" href="Categories.Adjoint.Equivalents.html#1835" class="Bound">adjoint</a>
<a id="1888" class="Comment">-- in this case, the hom functors are naturally isomorphism directly</a>
<a id="1961" href="Categories.Adjoint.Equivalents.html#1961" class="Function">Hom-NI′</a> <a id="1969" class="Symbol">:</a> <a id="1971" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a> <a id="1990" href="Categories.Adjoint.html#3049" class="Function">Hom[L-,-]</a> <a id="2000" href="Categories.Adjoint.html#3139" class="Function">Hom[-,R-]</a>
<a id="2014" href="Categories.Adjoint.Equivalents.html#1961" class="Function">Hom-NI′</a> <a id="2022" class="Symbol">=</a> <a id="2024" class="Keyword">record</a>
<a id="2037" class="Symbol">{</a> <a id="2039" href="Categories.NaturalTransformation.NaturalIsomorphism.html#891" class="Field">F⇒G</a> <a id="2043" class="Symbol">=</a> <a id="2045" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="2054" class="Keyword">record</a>
<a id="2069" class="Symbol">{</a> <a id="2071" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="2079" class="Symbol">=</a> <a id="2081" class="Symbol">λ</a> <a id="2083" href="Categories.Adjoint.Equivalents.html#2083" class="Bound">_</a> <a id="2085" class="Symbol">→</a> <a id="2087" class="Keyword">record</a> <a id="2094" class="Symbol">{</a> <a id="2096" href="Function.Bundles.html#2094" class="Field">to</a> <a id="2099" class="Symbol">=</a> <a id="2101" href="Function.Bundles.html#7394" class="Function">Hom-inverse.to</a> <a id="2116" class="Symbol">;</a> <a id="2118" href="Function.Bundles.html#2113" class="Field">cong</a> <a id="2123" class="Symbol">=</a> <a id="2125" href="Function.Bundles.html#7442" class="Function">Hom-inverse.to-cong</a> <a id="2145" class="Symbol">}</a>
<a id="2155" class="Symbol">;</a> <a id="2157" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="2165" class="Symbol">=</a> <a id="2167" class="Symbol">λ</a> <a id="2169" href="Categories.Adjoint.Equivalents.html#2169" class="Bound">_</a> <a id="2171" class="Symbol">→</a> <a id="2173" href="Categories.Adjoint.html#4238" class="Function">Ladjunct-comm</a> <a id="2187" href="Relation.Binary.Structures.html#1596" class="Function">D.Equiv.refl</a>
<a id="2208" class="Symbol">}</a>
<a id="2216" class="Symbol">;</a> <a id="2218" href="Categories.NaturalTransformation.NaturalIsomorphism.html#927" class="Field">F⇐G</a> <a id="2222" class="Symbol">=</a> <a id="2224" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="2233" class="Keyword">record</a>
<a id="2248" class="Symbol">{</a> <a id="2250" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="2258" class="Symbol">=</a> <a id="2260" class="Symbol">λ</a> <a id="2262" href="Categories.Adjoint.Equivalents.html#2262" class="Bound">_</a> <a id="2264" class="Symbol">→</a> <a id="2266" class="Keyword">record</a> <a id="2273" class="Symbol">{</a> <a id="2275" href="Function.Bundles.html#2094" class="Field">to</a> <a id="2278" class="Symbol">=</a> <a id="2280" href="Function.Bundles.html#7418" class="Function">Hom-inverse.from</a> <a id="2297" class="Symbol">;</a> <a id="2299" href="Function.Bundles.html#2113" class="Field">cong</a> <a id="2304" class="Symbol">=</a> <a id="2306" href="Function.Bundles.html#7483" class="Function">Hom-inverse.from-cong</a> <a id="2328" class="Symbol">}</a>
<a id="2338" class="Symbol">;</a> <a id="2340" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="2348" class="Symbol">=</a> <a id="2350" class="Symbol">λ</a> <a id="2352" href="Categories.Adjoint.Equivalents.html#2352" class="Bound">_</a> <a id="2354" class="Symbol">→</a> <a id="2356" href="Categories.Adjoint.html#5393" class="Function">Radjunct-comm</a> <a id="2370" href="Relation.Binary.Structures.html#1596" class="Function">C.Equiv.refl</a>
<a id="2391" class="Symbol">}</a>
<a id="2399" class="Symbol">;</a> <a id="2401" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1051" class="Field">iso</a> <a id="2405" class="Symbol">=</a> <a id="2407" class="Symbol">λ</a> <a id="2409" href="Categories.Adjoint.Equivalents.html#2409" class="Bound">_</a> <a id="2411" class="Symbol">→</a> <a id="2413" class="Keyword">record</a>
<a id="2428" class="Symbol">{</a> <a id="2430" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="2435" class="Symbol">=</a> <a id="2437" href="Categories.Adjoint.html#2026" class="Function">RLadjunct≈id</a>
<a id="2458" class="Symbol">;</a> <a id="2460" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="2465" class="Symbol">=</a> <a id="2467" href="Categories.Adjoint.html#2533" class="Function">LRadjunct≈id</a>
<a id="2488" class="Symbol">}</a>
<a id="2496" class="Symbol">}</a>
<a id="2501" class="Comment">-- now goes from natural isomorphism back to adjoint.</a>
<a id="2557" class="Comment">-- for simplicity, just construct the case in which homsetoids of C and D</a>
<a id="2633" class="Comment">-- are compatible.</a>
<a id="2655" class="Keyword">private</a>
<a id="2667" href="Categories.Adjoint.Equivalents.html#2667" class="Function">Hom[L-,-]</a> <a id="2677" class="Symbol">:</a> <a id="2679" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="2689" href="Categories.Category.Core.html#3132" class="Function">C.op</a> <a id="2694" href="Categories.Adjoint.Equivalents.html#1647" class="Bound">D</a> <a id="2696" class="Symbol">(</a><a id="2697" href="Categories.Category.Instance.Setoids.html#555" class="Function">Setoids</a> <a id="2705" class="Symbol">_</a> <a id="2707" class="Symbol">_)</a>
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<a id="5340" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5342" href="Categories.Adjoint.Equivalents.html#4176" class="Function">counit</a> <a id="5349" class="Symbol">(</a><a id="5350" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="5355" href="Categories.Adjoint.Equivalents.html#5214" class="Bound">A</a><a id="5356" class="Symbol">)</a> <a id="5358" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5360" href="Categories.Functor.Core.html#455" class="Function">L.F₁</a> <a id="5365" class="Symbol">(</a><a id="5366" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5368" href="Categories.Adjoint.Equivalents.html#3140" class="Function">unit</a> <a id="5373" href="Categories.Adjoint.Equivalents.html#5214" class="Bound">A</a><a id="5374" class="Symbol">)</a> <a id="5381" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="5385" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a> <a id="5395" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="5407" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="5410" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5412" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5414" href="Categories.Adjoint.Equivalents.html#4176" class="Function">counit</a> <a id="5421" class="Symbol">(</a><a id="5422" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="5427" href="Categories.Adjoint.Equivalents.html#5214" class="Bound">A</a><a id="5428" class="Symbol">)</a> <a id="5430" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5432" href="Categories.Functor.Core.html#455" class="Function">L.F₁</a> <a id="5437" class="Symbol">(</a><a id="5438" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5440" href="Categories.Adjoint.Equivalents.html#3140" class="Function">unit</a> <a id="5445" href="Categories.Adjoint.Equivalents.html#5214" class="Bound">A</a><a id="5446" class="Symbol">)</a> <a id="5448" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="5452" href="Categories.NaturalTransformation.Core.html#827" class="Function">⇐.commute</a> <a id="5462" class="Symbol">(</a><a id="5463" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5465" href="Categories.Adjoint.Equivalents.html#3140" class="Function">unit</a> <a id="5470" href="Categories.Adjoint.Equivalents.html#5214" class="Bound">A</a> <a id="5472" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5474" href="Categories.Category.Core.html#630" class="Function">id</a><a id="5476" class="Symbol">)</a> <a id="5478" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="5490" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="5494" class="Symbol">(</a><a id="5495" href="Categories.Adjoint.Equivalents.html#5214" class="Bound">A</a> <a id="5497" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5499" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="5504" href="Categories.Adjoint.Equivalents.html#5214" class="Bound">A</a><a id="5505" class="Symbol">)</a> <a id="5507" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="5511" class="Symbol">(</a><a id="5512" href="Categories.Functor.Core.html#455" class="Field">R.F₁</a> <a id="5517" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="5520" href="Categories.Category.Core.html#656" class="Function Operator">C.∘</a> <a id="5524" href="Categories.Category.Core.html#630" class="Function">C.id</a> <a id="5529" href="Categories.Category.Core.html#656" class="Function Operator">C.∘</a> <a id="5533" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5535" href="Categories.Adjoint.Equivalents.html#3140" class="Function">unit</a> <a id="5540" href="Categories.Adjoint.Equivalents.html#5214" class="Bound">A</a><a id="5541" class="Symbol">)</a>
<a id="5594" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5597" href="Function.Bundles.html#2113" class="Field">cong</a> <a id="5602" class="Symbol">(</a><a id="5603" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="5607" class="Symbol">(</a><a id="5608" href="Categories.Adjoint.Equivalents.html#5214" class="Bound">A</a> <a id="5610" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5612" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="5617" href="Categories.Adjoint.Equivalents.html#5214" class="Bound">A</a><a id="5618" class="Symbol">))</a> <a id="5621" class="Symbol">(</a><a id="5622" href="Relation.Binary.Structures.html#1648" class="Function">C.Equiv.trans</a> <a id="5636" class="Symbol">(</a><a id="5637" href="Categories.Morphism.Reasoning.Core.html#2948" class="Function">MR.elimˡ</a> <a id="5646" href="Categories.Adjoint.Equivalents.html#1626" class="Bound">C</a> <a id="5648" href="Categories.Functor.Core.html#511" class="Field">R.identity</a><a id="5658" class="Symbol">)</a> <a id="5660" href="Categories.Category.Core.html#1096" class="Function">C.identityˡ</a><a id="5671" class="Symbol">)</a> <a id="5673" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5685" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="5689" class="Symbol">(</a><a id="5690" href="Categories.Adjoint.Equivalents.html#5214" class="Bound">A</a> <a id="5692" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5694" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="5699" href="Categories.Adjoint.Equivalents.html#5214" class="Bound">A</a><a id="5700" class="Symbol">)</a> <a id="5702" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="5706" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5708" href="Categories.Adjoint.Equivalents.html#3140" class="Function">unit</a> <a id="5713" href="Categories.Adjoint.Equivalents.html#5214" class="Bound">A</a> <a id="5726" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5729" href="Categories.Morphism.html#1586" class="Function">isoˡ</a> <a id="5734" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5746" href="Categories.Category.Core.html#630" class="Function">id</a>
<a id="5800" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="5808" class="Symbol">;</a> <a id="5810" href="Categories.Adjoint.html#1742" class="Field">zag</a> <a id="5817" class="Symbol">=</a> <a id="5819" class="Symbol">λ</a> <a id="5821" class="Symbol">{</a><a id="5822" href="Categories.Adjoint.Equivalents.html#5822" class="Bound">B</a><a id="5823" class="Symbol">}</a> <a id="5825" class="Symbol">→</a>
<a id="5835" class="Keyword">let</a> <a id="5839" class="Keyword">open</a> <a id="5844" href="Categories.Adjoint.Equivalents.html#1731" class="Module">C</a>
<a id="5858" class="Keyword">open</a> <a id="5863" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="5888" class="Keyword">open</a> <a id="5893" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="5911" class="Keyword">open</a> <a id="5916" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="5919" href="Categories.Adjoint.Equivalents.html#1626" class="Bound">C</a>
<a id="5929" class="Keyword">in</a> <a id="5932" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="5948" href="Categories.Functor.Core.html#455" class="Field">R.F₁</a> <a id="5953" class="Symbol">(</a><a id="5954" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5956" href="Categories.Adjoint.Equivalents.html#4176" class="Function">counit</a> <a id="5963" href="Categories.Adjoint.Equivalents.html#5822" class="Bound">B</a><a id="5964" class="Symbol">)</a> <a id="5966" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5968" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5970" href="Categories.Adjoint.Equivalents.html#3140" class="Function">unit</a> <a id="5975" class="Symbol">(</a><a id="5976" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="5981" href="Categories.Adjoint.Equivalents.html#5822" class="Bound">B</a><a id="5982" class="Symbol">)</a> <a id="5989" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="5993" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="6001" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a> <a id="6011" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="6023" href="Categories.Functor.Core.html#455" class="Field">R.F₁</a> <a id="6028" class="Symbol">(</a><a id="6029" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="6031" href="Categories.Adjoint.Equivalents.html#4176" class="Function">counit</a> <a id="6038" href="Categories.Adjoint.Equivalents.html#5822" class="Bound">B</a><a id="6039" class="Symbol">)</a> <a id="6041" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6043" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="6045" href="Categories.Adjoint.Equivalents.html#3140" class="Function">unit</a> <a id="6050" class="Symbol">(</a><a id="6051" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="6056" href="Categories.Adjoint.Equivalents.html#5822" class="Bound">B</a><a id="6057" class="Symbol">)</a> <a id="6059" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6061" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6064" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="6068" href="Categories.NaturalTransformation.Core.html#827" class="Function">⇒.commute</a> <a id="6078" class="Symbol">(</a><a id="6079" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6082" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="6084" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="6086" href="Categories.Adjoint.Equivalents.html#4176" class="Function">counit</a> <a id="6093" href="Categories.Adjoint.Equivalents.html#5822" class="Bound">B</a><a id="6094" class="Symbol">)</a> <a id="6096" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="6108" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="6112" class="Symbol">(</a><a id="6113" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="6118" href="Categories.Adjoint.Equivalents.html#5822" class="Bound">B</a> <a id="6120" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="6122" href="Categories.Adjoint.Equivalents.html#5822" class="Bound">B</a><a id="6123" class="Symbol">)</a> <a id="6125" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="6129" class="Symbol">(</a><a id="6130" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="6132" href="Categories.Adjoint.Equivalents.html#4176" class="Function">counit</a> <a id="6139" href="Categories.Adjoint.Equivalents.html#5822" class="Bound">B</a> <a id="6141" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="6145" href="Categories.Category.Core.html#630" class="Function">D.id</a> <a id="6150" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="6154" href="Categories.Functor.Core.html#455" class="Function">L.F₁</a> <a id="6159" href="Categories.Category.Core.html#630" class="Function">id</a><a id="6161" class="Symbol">)</a>
<a id="6214" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6217" href="Function.Bundles.html#2113" class="Field">cong</a> <a id="6222" class="Symbol">(</a><a id="6223" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="6227" class="Symbol">(</a><a id="6228" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="6233" href="Categories.Adjoint.Equivalents.html#5822" class="Bound">B</a> <a id="6235" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="6237" href="Categories.Adjoint.Equivalents.html#5822" class="Bound">B</a><a id="6238" class="Symbol">))</a> <a id="6241" class="Symbol">(</a><a id="6242" href="Categories.Morphism.Reasoning.Core.html#2786" class="Function">MR.elimʳ</a> <a id="6251" href="Categories.Adjoint.Equivalents.html#1647" class="Bound">D</a> <a id="6253" class="Symbol">(</a><a id="6254" href="Categories.Morphism.Reasoning.Core.html#2786" class="Function">MR.elimʳ</a> <a id="6263" href="Categories.Adjoint.Equivalents.html#1647" class="Bound">D</a> <a id="6265" href="Categories.Functor.Core.html#511" class="Function">L.identity</a><a id="6275" class="Symbol">))</a> <a id="6278" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6290" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="6294" class="Symbol">(</a><a id="6295" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="6300" href="Categories.Adjoint.Equivalents.html#5822" class="Bound">B</a> <a id="6302" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="6304" href="Categories.Adjoint.Equivalents.html#5822" class="Bound">B</a><a id="6305" class="Symbol">)</a> <a id="6307" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="6311" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="6313" href="Categories.Adjoint.Equivalents.html#4176" class="Function">counit</a> <a id="6320" href="Categories.Adjoint.Equivalents.html#5822" class="Bound">B</a> <a id="6331" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6334" href="Categories.Morphism.html#1612" class="Function">isoʳ</a> <a id="6339" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6351" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6392" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="6400" class="Symbol">}</a>
<a id="6408" class="Keyword">where</a> <a id="6414" class="Keyword">module</a> <a id="6421" href="Categories.Adjoint.Equivalents.html#6421" class="Module">i</a> <a id="6423" class="Symbol">{</a><a id="6424" href="Categories.Adjoint.Equivalents.html#6424" class="Bound">X</a><a id="6425" class="Symbol">}</a> <a id="6427" class="Symbol">=</a> <a id="6429" href="Categories.Morphism.html#1528" class="Module">Iso</a> <a id="6433" class="Symbol">(</a><a id="6434" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1051" class="Field">iso</a> <a id="6438" href="Categories.Adjoint.Equivalents.html#6424" class="Bound">X</a><a id="6439" class="Symbol">)</a>
<a id="6453" class="Keyword">open</a> <a id="6458" href="Categories.Adjoint.Equivalents.html#6421" class="Module">i</a>
<a id="6461" class="Comment">-- the general case from isomorphic Hom setoids to adjoint functors</a>
<a id="6529" class="Keyword">module</a> <a id="6536" href="Categories.Adjoint.Equivalents.html#6536" class="Module">_</a> <a id="6538" class="Symbol">{</a><a id="6539" href="Categories.Adjoint.Equivalents.html#6539" class="Bound">C</a> <a id="6541" class="Symbol">:</a> <a id="6543" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="6552" href="Categories.Adjoint.Equivalents.html#1335" class="Generalizable">o</a> <a id="6554" href="Categories.Adjoint.Equivalents.html#1343" class="Generalizable">ℓ</a> <a id="6556" href="Categories.Adjoint.Equivalents.html#1351" class="Generalizable">e</a><a id="6557" class="Symbol">}</a> <a id="6559" class="Symbol">{</a><a id="6560" href="Categories.Adjoint.Equivalents.html#6560" class="Bound">D</a> <a id="6562" class="Symbol">:</a> <a id="6564" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="6573" href="Categories.Adjoint.Equivalents.html#1337" class="Generalizable">o′</a> <a id="6576" href="Categories.Adjoint.Equivalents.html#1345" class="Generalizable">ℓ′</a> <a id="6579" href="Categories.Adjoint.Equivalents.html#1353" class="Generalizable">e′</a><a id="6581" class="Symbol">}</a> <a id="6583" class="Symbol">{</a><a id="6584" href="Categories.Adjoint.Equivalents.html#6584" class="Bound">L</a> <a id="6586" class="Symbol">:</a> <a id="6588" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="6596" href="Categories.Adjoint.Equivalents.html#6539" class="Bound">C</a> <a id="6598" href="Categories.Adjoint.Equivalents.html#6560" class="Bound">D</a><a id="6599" class="Symbol">}</a> <a id="6601" class="Symbol">{</a><a id="6602" href="Categories.Adjoint.Equivalents.html#6602" class="Bound">R</a> <a id="6604" class="Symbol">:</a> <a id="6606" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="6614" href="Categories.Adjoint.Equivalents.html#6560" class="Bound">D</a> <a id="6616" href="Categories.Adjoint.Equivalents.html#6539" class="Bound">C</a><a id="6617" class="Symbol">}</a> <a id="6619" class="Keyword">where</a>
<a id="6627" class="Keyword">private</a>
<a id="6639" class="Keyword">module</a> <a id="6646" href="Categories.Adjoint.Equivalents.html#6646" class="Module">C</a> <a id="6648" class="Symbol">=</a> <a id="6650" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="6659" href="Categories.Adjoint.Equivalents.html#6539" class="Bound">C</a>
<a id="6665" class="Keyword">module</a> <a id="6672" href="Categories.Adjoint.Equivalents.html#6672" class="Module">D</a> <a id="6674" class="Symbol">=</a> <a id="6676" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="6685" href="Categories.Adjoint.Equivalents.html#6560" class="Bound">D</a>
<a id="6691" class="Keyword">module</a> <a id="6698" href="Categories.Adjoint.Equivalents.html#6698" class="Module">L</a> <a id="6700" class="Symbol">=</a> <a id="6702" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="6710" href="Categories.Adjoint.Equivalents.html#6584" class="Bound">L</a>
<a id="6716" class="Keyword">module</a> <a id="6723" href="Categories.Adjoint.Equivalents.html#6723" class="Module">R</a> <a id="6725" class="Symbol">=</a> <a id="6727" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="6735" href="Categories.Adjoint.Equivalents.html#6602" class="Bound">R</a>
<a id="6741" class="Keyword">open</a> <a id="6746" href="Categories.Functor.Core.html#248" class="Module">Functor</a>
<a id="6758" class="Keyword">open</a> <a id="6763" href="Function.Bundles.html#2043" class="Module">Func</a>
<a id="6773" href="Categories.Adjoint.Equivalents.html#6773" class="Function">Hom[L-,-]</a> <a id="6783" class="Symbol">:</a> <a id="6785" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="6795" href="Categories.Category.Core.html#3132" class="Function">C.op</a> <a id="6800" href="Categories.Adjoint.Equivalents.html#6560" class="Bound">D</a> <a id="6802" class="Symbol">(</a><a id="6803" href="Categories.Category.Instance.Setoids.html#555" class="Function">Setoids</a> <a id="6811" class="Symbol">_</a> <a id="6813" class="Symbol">_)</a>
<a id="6820" href="Categories.Adjoint.Equivalents.html#6773" class="Function">Hom[L-,-]</a> <a id="6830" class="Symbol">=</a> <a id="6832" href="Categories.Functor.Construction.LiftSetoids.html#1046" class="Function">LiftSetoids</a> <a id="6844" href="Categories.Adjoint.Equivalents.html#6554" class="Bound">ℓ</a> <a id="6846" href="Categories.Adjoint.Equivalents.html#6556" class="Bound">e</a> <a id="6848" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="6851" href="Categories.Functor.Hom.html#1578" class="Function Operator">Hom[</a> <a id="6856" href="Categories.Adjoint.Equivalents.html#6560" class="Bound">D</a> <a id="6858" href="Categories.Functor.Hom.html#1578" class="Function Operator">][-,-]</a> <a id="6865" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="6868" class="Symbol">(</a><a id="6869" href="Categories.Functor.Core.html#816" class="Function">L.op</a> <a id="6874" href="Categories.Category.Product.html#1962" class="Function Operator">⁂</a> <a id="6876" href="Categories.Adjoint.Equivalents.html#833" class="Function">idF</a><a id="6879" class="Symbol">)</a>
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<a id="10687" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="10691" href="Level.html#479" class="Field">lower</a> <a id="10697" class="Symbol">(</a><a id="10698" href="Categories.NaturalTransformation.Core.html#827" class="Function">⇒.commute</a> <a id="10708" class="Symbol">(</a><a id="10709" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="10712" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="10714" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="10716" href="Categories.Adjoint.Equivalents.html#8415" class="Function">counit</a> <a id="10723" href="Categories.Adjoint.Equivalents.html#10304" class="Bound">B</a><a id="10724" class="Symbol">))</a> <a id="10727" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="10739" href="Level.html#479" class="Field">lower</a> <a id="10745" class="Symbol">(</a><a id="10746" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="10750" class="Symbol">(</a><a id="10751" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="10756" href="Categories.Adjoint.Equivalents.html#10304" class="Bound">B</a> <a id="10758" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="10760" href="Categories.Adjoint.Equivalents.html#10304" class="Bound">B</a><a id="10761" class="Symbol">)</a> <a id="10763" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="10767" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="10772" class="Symbol">(</a><a id="10773" href="Level.html#479" class="Field">lower</a> <a id="10779" class="Symbol">(</a><a id="10780" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="10784" class="Symbol">(</a><a id="10785" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="10790" href="Categories.Adjoint.Equivalents.html#10304" class="Bound">B</a> <a id="10792" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="10794" href="Categories.Adjoint.Equivalents.html#10304" class="Bound">B</a><a id="10795" class="Symbol">)</a> <a id="10797" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="10801" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="10806" href="Categories.Category.Core.html#630" class="Function">id</a><a id="10808" class="Symbol">)</a> <a id="10810" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="10814" href="Categories.Category.Core.html#630" class="Function">D.id</a> <a id="10819" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="10823" href="Categories.Functor.Core.html#455" class="Function">L.F₁</a> <a id="10828" href="Categories.Category.Core.html#630" class="Function">id</a><a id="10830" class="Symbol">))</a>
<a id="10845" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10848" href="Level.html#479" class="Field">lower</a> <a id="10854" class="Symbol">(</a><a id="10855" href="Function.Bundles.html#2113" class="Field">cong</a> <a id="10860" class="Symbol">(</a><a id="10861" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="10865" class="Symbol">(</a><a id="10866" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="10871" href="Categories.Adjoint.Equivalents.html#10304" class="Bound">B</a> <a id="10873" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="10875" href="Categories.Adjoint.Equivalents.html#10304" class="Bound">B</a><a id="10876" class="Symbol">))</a> <a id="10879" class="Symbol">(</a><a id="10880" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="10885" class="Symbol">(</a><a id="10886" href="Categories.Morphism.Reasoning.Core.html#2786" class="Function">MR.elimʳ</a> <a id="10895" href="Categories.Adjoint.Equivalents.html#6560" class="Bound">D</a> <a id="10897" class="Symbol">(</a><a id="10898" href="Categories.Morphism.Reasoning.Core.html#2786" class="Function">MR.elimʳ</a> <a id="10907" href="Categories.Adjoint.Equivalents.html#6560" class="Bound">D</a> <a id="10909" href="Categories.Functor.Core.html#511" class="Function">L.identity</a><a id="10919" class="Symbol">))))</a> <a id="10924" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10936" href="Level.html#479" class="Field">lower</a> <a id="10942" class="Symbol">(</a><a id="10943" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="10947" class="Symbol">(</a><a id="10948" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="10953" href="Categories.Adjoint.Equivalents.html#10304" class="Bound">B</a> <a id="10955" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="10957" href="Categories.Adjoint.Equivalents.html#10304" class="Bound">B</a><a id="10958" class="Symbol">)</a> <a id="10960" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="10964" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="10969" class="Symbol">(</a><a id="10970" href="Level.html#479" class="Field">lower</a> <a id="10976" class="Symbol">(</a><a id="10977" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="10981" class="Symbol">(</a><a id="10982" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="10987" href="Categories.Adjoint.Equivalents.html#10304" class="Bound">B</a> <a id="10989" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="10991" href="Categories.Adjoint.Equivalents.html#10304" class="Bound">B</a><a id="10992" class="Symbol">)</a> <a id="10994" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="10998" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="11003" href="Categories.Category.Core.html#630" class="Function">id</a><a id="11005" class="Symbol">)))</a>
<a id="11021" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="11024" href="Level.html#479" class="Field">lower</a> <a id="11030" href="Categories.Morphism.html#1612" class="Function">isoʳ</a> <a id="11035" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="11047" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="11050" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="11058" class="Symbol">}</a>
<a id="11066" class="Keyword">where</a> <a id="11072" class="Keyword">open</a> <a id="11077" href="Categories.NaturalTransformation.Core.html#466" class="Module">NaturalTransformation</a>
<a id="11111" class="Keyword">module</a> <a id="11118" href="Categories.Adjoint.Equivalents.html#11118" class="Module">_</a> <a id="11120" class="Symbol">{</a><a id="11121" href="Categories.Adjoint.Equivalents.html#11121" class="Bound">X</a><a id="11122" class="Symbol">}</a> <a id="11124" class="Keyword">where</a>
<a id="11144" class="Keyword">open</a> <a id="11149" href="Categories.Morphism.html#1528" class="Module">Iso</a> <a id="11153" class="Symbol">(</a><a id="11154" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1051" class="Field">iso</a> <a id="11158" href="Categories.Adjoint.Equivalents.html#11121" class="Bound">X</a><a id="11159" class="Symbol">)</a> <a id="11161" class="Keyword">public</a>
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