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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#218" class="InductiveConstructor Operator">_,_</a><a id="219" class="Symbol">;</a> <a id="221" href="Data.Product.html#1176" 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#1928" 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#1040" 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.Equality.html" class="Module">Function.Equality</a> <a id="311" class="Keyword">using</a> <a id="317" class="Symbol">(</a><a id="318" href="Function.Equality.html#898" class="Record">Π</a><a id="319" class="Symbol">;</a> <a id="321" href="Function.Equality.html#1227" class="Function Operator">_⟶_</a><a id="324" class="Symbol">)</a>
<a id="326" class="Keyword">import</a> <a id="333" href="Function.Inverse.html" class="Module">Function.Inverse</a> <a id="350" class="Symbol">as</a> <a id="353" class="Module">FI</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#891" class="Function">Rel</a><a id="394" class="Symbol">;</a> <a id="396" href="Relation.Binary.Structures.html#1531" class="Record">IsEquivalence</a><a id="409" class="Symbol">;</a> <a id="411" href="Relation.Binary.Bundles.html#1018" 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#1306" class="Record">Adjoint</a><a id="533" class="Symbol">;</a> <a id="535" href="Categories.Adjoint.html#7972" 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">open</a> <a id="709" class="Keyword">import</a> <a id="716" href="Categories.Morphism.html" class="Module">Categories.Morphism</a>
<a id="736" class="Keyword">open</a> <a id="741" class="Keyword">import</a> <a id="748" href="Categories.Functor.html" class="Module">Categories.Functor</a> <a id="767" class="Keyword">using</a> <a id="773" class="Symbol">(</a><a id="774" href="Categories.Functor.Core.html#248" class="Record">Functor</a><a id="781" class="Symbol">;</a> <a id="783" href="Categories.Functor.html#747" class="Function Operator">_∘F_</a><a id="787" class="Symbol">)</a> <a id="789" class="Keyword">renaming</a> <a id="798" class="Symbol">(</a><a id="799" href="Categories.Functor.html#349" class="Function">id</a> <a id="802" class="Symbol">to</a> <a id="805" class="Function">idF</a><a id="808" class="Symbol">)</a>
<a id="810" class="Keyword">open</a> <a id="815" class="Keyword">import</a> <a id="822" href="Categories.Functor.Bifunctor.html" class="Module">Categories.Functor.Bifunctor</a> <a id="851" class="Keyword">using</a> <a id="857" class="Symbol">(</a><a id="858" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a><a id="867" class="Symbol">)</a>
<a id="869" class="Keyword">open</a> <a id="874" class="Keyword">import</a> <a id="881" href="Categories.Functor.Hom.html" class="Module">Categories.Functor.Hom</a> <a id="904" class="Keyword">using</a> <a id="910" class="Symbol">(</a><a id="911" href="Categories.Functor.Hom.html#2973" class="Function Operator">Hom[_][-,-]</a><a id="922" class="Symbol">)</a>
<a id="924" class="Keyword">open</a> <a id="929" class="Keyword">import</a> <a id="936" href="Categories.Functor.Construction.LiftSetoids.html" class="Module">Categories.Functor.Construction.LiftSetoids</a>
<a id="980" class="Keyword">open</a> <a id="985" class="Keyword">import</a> <a id="992" href="Categories.NaturalTransformation.html" class="Module">Categories.NaturalTransformation</a> <a id="1025" class="Keyword">using</a> <a id="1031" class="Symbol">(</a><a id="1032" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a><a id="1053" class="Symbol">;</a> <a id="1055" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a><a id="1063" class="Symbol">;</a> <a id="1065" href="Categories.NaturalTransformation.Core.html#2919" class="Function Operator">_∘ₕ_</a><a id="1069" class="Symbol">;</a> <a id="1071" href="Categories.NaturalTransformation.Core.html#2439" class="Function Operator">_∘ᵥ_</a><a id="1075" class="Symbol">;</a> <a id="1077" href="Categories.NaturalTransformation.Core.html#3439" class="Function Operator">_∘ˡ_</a><a id="1081" class="Symbol">;</a> <a id="1083" href="Categories.NaturalTransformation.Core.html#3784" class="Function Operator">_∘ʳ_</a><a id="1087" class="Symbol">)</a>
<a id="1091" class="Keyword">renaming</a> <a id="1100" class="Symbol">(</a><a id="1101" href="Categories.NaturalTransformation.Core.html#2132" class="Function">id</a> <a id="1104" class="Symbol">to</a> <a id="1107" class="Function">idN</a><a id="1110" class="Symbol">)</a>
<a id="1112" class="Keyword">open</a> <a id="1117" class="Keyword">import</a> <a id="1124" href="Categories.NaturalTransformation.NaturalIsomorphism.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism</a>
<a id="1178" class="Keyword">using</a> <a id="1184" class="Symbol">(</a><a id="1185" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a><a id="1203" class="Symbol">;</a> <a id="1205" href="Categories.NaturalTransformation.NaturalIsomorphism.html#6216" class="Function">unitorˡ</a><a id="1212" class="Symbol">;</a> <a id="1214" href="Categories.NaturalTransformation.NaturalIsomorphism.html#6312" class="Function">unitorʳ</a><a id="1221" class="Symbol">;</a> <a id="1223" href="Categories.NaturalTransformation.NaturalIsomorphism.html#7073" class="Function">associator</a><a id="1233" class="Symbol">;</a> <a id="1235" href="Categories.NaturalTransformation.NaturalIsomorphism.html#3600" class="Function Operator">_≃_</a><a id="1238" class="Symbol">)</a>
<a id="1240" class="Keyword">import</a> <a id="1247" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="1277" class="Symbol">as</a> <a id="1280" class="Module">MR</a>
<a id="1284" class="Keyword">private</a>
<a id="1294" class="Keyword">variable</a>
<a id="1307" href="Categories.Adjoint.Equivalents.html#1307" class="Generalizable">o</a> <a id="1309" href="Categories.Adjoint.Equivalents.html#1309" class="Generalizable">o′</a> <a id="1312" href="Categories.Adjoint.Equivalents.html#1312" class="Generalizable">o″</a> <a id="1315" href="Categories.Adjoint.Equivalents.html#1315" class="Generalizable">ℓ</a> <a id="1317" href="Categories.Adjoint.Equivalents.html#1317" class="Generalizable">ℓ′</a> <a id="1320" href="Categories.Adjoint.Equivalents.html#1320" class="Generalizable">ℓ″</a> <a id="1323" href="Categories.Adjoint.Equivalents.html#1323" class="Generalizable">e</a> <a id="1325" href="Categories.Adjoint.Equivalents.html#1325" class="Generalizable">e′</a> <a id="1328" href="Categories.Adjoint.Equivalents.html#1328" class="Generalizable">e″</a> <a id="1331" class="Symbol">:</a> <a id="1333" href="Agda.Primitive.html#591" class="Postulate">Level</a>
<a id="1343" href="Categories.Adjoint.Equivalents.html#1343" class="Generalizable">C</a> <a id="1345" href="Categories.Adjoint.Equivalents.html#1345" class="Generalizable">D</a> <a id="1347" href="Categories.Adjoint.Equivalents.html#1347" class="Generalizable">E</a> <a id="1349" class="Symbol">:</a> <a id="1351" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="1360" href="Categories.Adjoint.Equivalents.html#1307" class="Generalizable">o</a> <a id="1362" href="Categories.Adjoint.Equivalents.html#1315" class="Generalizable">ℓ</a> <a id="1364" href="Categories.Adjoint.Equivalents.html#1323" class="Generalizable">e</a>
<a id="1367" class="Comment">-- a special case of the natural isomorphism in which homsets in C and D have the same</a>
<a id="1454" class="Comment">-- universe level. therefore there is no need to lift Setoids to the same level.</a>
<a id="1535" class="Comment">-- this is helpful when combining with Yoneda lemma.</a>
<a id="1588" class="Keyword">module</a> <a id="1595" href="Categories.Adjoint.Equivalents.html#1595" class="Module">_</a> <a id="1597" class="Symbol">{</a><a id="1598" href="Categories.Adjoint.Equivalents.html#1598" class="Bound">C</a> <a id="1600" class="Symbol">:</a> <a id="1602" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="1611" href="Categories.Adjoint.Equivalents.html#1307" class="Generalizable">o</a> <a id="1613" href="Categories.Adjoint.Equivalents.html#1315" class="Generalizable">ℓ</a> <a id="1615" href="Categories.Adjoint.Equivalents.html#1323" class="Generalizable">e</a><a id="1616" class="Symbol">}</a> <a id="1618" class="Symbol">{</a><a id="1619" href="Categories.Adjoint.Equivalents.html#1619" class="Bound">D</a> <a id="1621" class="Symbol">:</a> <a id="1623" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="1632" href="Categories.Adjoint.Equivalents.html#1309" class="Generalizable">o′</a> <a id="1635" href="Categories.Adjoint.Equivalents.html#1315" class="Generalizable">ℓ</a> <a id="1637" href="Categories.Adjoint.Equivalents.html#1323" class="Generalizable">e</a><a id="1638" class="Symbol">}</a> <a id="1640" class="Symbol">{</a><a id="1641" href="Categories.Adjoint.Equivalents.html#1641" class="Bound">L</a> <a id="1643" class="Symbol">:</a> <a id="1645" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="1653" href="Categories.Adjoint.Equivalents.html#1598" class="Bound">C</a> <a id="1655" href="Categories.Adjoint.Equivalents.html#1619" class="Bound">D</a><a id="1656" class="Symbol">}</a> <a id="1658" class="Symbol">{</a><a id="1659" href="Categories.Adjoint.Equivalents.html#1659" class="Bound">R</a> <a id="1661" class="Symbol">:</a> <a id="1663" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="1671" href="Categories.Adjoint.Equivalents.html#1619" class="Bound">D</a> <a id="1673" href="Categories.Adjoint.Equivalents.html#1598" class="Bound">C</a><a id="1674" class="Symbol">}</a> <a id="1676" class="Keyword">where</a>
<a id="1684" class="Keyword">private</a>
<a id="1696" class="Keyword">module</a> <a id="1703" href="Categories.Adjoint.Equivalents.html#1703" class="Module">C</a> <a id="1705" class="Symbol">=</a> <a id="1707" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1716" href="Categories.Adjoint.Equivalents.html#1598" class="Bound">C</a>
<a id="1722" class="Keyword">module</a> <a id="1729" href="Categories.Adjoint.Equivalents.html#1729" class="Module">D</a> <a id="1731" class="Symbol">=</a> <a id="1733" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1742" href="Categories.Adjoint.Equivalents.html#1619" class="Bound">D</a>
<a id="1748" class="Keyword">module</a> <a id="1755" href="Categories.Adjoint.Equivalents.html#1755" class="Module">L</a> <a id="1757" class="Symbol">=</a> <a id="1759" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1767" href="Categories.Adjoint.Equivalents.html#1641" class="Bound">L</a>
<a id="1773" class="Keyword">module</a> <a id="1780" href="Categories.Adjoint.Equivalents.html#1780" class="Module">R</a> <a id="1782" class="Symbol">=</a> <a id="1784" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1792" href="Categories.Adjoint.Equivalents.html#1659" class="Bound">R</a>
<a id="1797" class="Keyword">module</a> <a id="1804" href="Categories.Adjoint.Equivalents.html#1804" class="Module">_</a> <a id="1806" class="Symbol">(</a><a id="1807" href="Categories.Adjoint.Equivalents.html#1807" class="Bound">adjoint</a> <a id="1815" class="Symbol">:</a> <a id="1817" href="Categories.Adjoint.Equivalents.html#1641" class="Bound">L</a> <a id="1819" href="Categories.Adjoint.html#7972" class="Function Operator">⊣</a> <a id="1821" href="Categories.Adjoint.Equivalents.html#1659" class="Bound">R</a><a id="1822" class="Symbol">)</a> <a id="1824" class="Keyword">where</a>
<a id="1834" class="Keyword">open</a> <a id="1839" href="Categories.Adjoint.html#1306" class="Module">Adjoint</a> <a id="1847" href="Categories.Adjoint.Equivalents.html#1807" class="Bound">adjoint</a>
<a id="1860" class="Comment">-- in this case, the hom functors are naturally isomorphism directly</a>
<a id="1933" href="Categories.Adjoint.Equivalents.html#1933" class="Function">Hom-NI′</a> <a id="1941" class="Symbol">:</a> <a id="1943" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a> <a id="1962" href="Categories.Adjoint.html#3095" class="Function">Hom[L-,-]</a> <a id="1972" href="Categories.Adjoint.html#3185" class="Function">Hom[-,R-]</a>
<a id="1986" href="Categories.Adjoint.Equivalents.html#1933" class="Function">Hom-NI′</a> <a id="1994" class="Symbol">=</a> <a id="1996" class="Keyword">record</a>
<a id="2009" class="Symbol">{</a> <a id="2011" href="Categories.NaturalTransformation.NaturalIsomorphism.html#891" class="Field">F⇒G</a> <a id="2015" class="Symbol">=</a> <a id="2017" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="2026" class="Keyword">record</a>
<a id="2041" class="Symbol">{</a> <a id="2043" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="2051" class="Symbol">=</a> <a id="2053" class="Symbol">λ</a> <a id="2055" href="Categories.Adjoint.Equivalents.html#2055" class="Bound">_</a> <a id="2057" class="Symbol">→</a> <a id="2059" href="Function.Inverse.html#1532" class="Function">Hom-inverse.to</a>
<a id="2082" class="Symbol">;</a> <a id="2084" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="2092" class="Symbol">=</a> <a id="2094" class="Symbol">λ</a> <a id="2096" href="Categories.Adjoint.Equivalents.html#2096" class="Bound">_</a> <a id="2098" href="Categories.Adjoint.Equivalents.html#2098" class="Bound">eq</a> <a id="2101" class="Symbol">→</a> <a id="2103" href="Categories.Adjoint.html#4288" class="Function">Ladjunct-comm</a> <a id="2117" href="Categories.Adjoint.Equivalents.html#2098" class="Bound">eq</a>
<a id="2128" class="Symbol">}</a>
<a id="2136" class="Symbol">;</a> <a id="2138" href="Categories.NaturalTransformation.NaturalIsomorphism.html#927" class="Field">F⇐G</a> <a id="2142" class="Symbol">=</a> <a id="2144" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="2153" class="Keyword">record</a>
<a id="2168" class="Symbol">{</a> <a id="2170" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="2178" class="Symbol">=</a> <a id="2180" class="Symbol">λ</a> <a id="2182" href="Categories.Adjoint.Equivalents.html#2182" class="Bound">_</a> <a id="2184" class="Symbol">→</a> <a id="2186" href="Function.Inverse.html#1559" class="Function">Hom-inverse.from</a>
<a id="2211" class="Symbol">;</a> <a id="2213" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="2221" class="Symbol">=</a> <a id="2223" class="Symbol">λ</a> <a id="2225" href="Categories.Adjoint.Equivalents.html#2225" class="Bound">_</a> <a id="2227" href="Categories.Adjoint.Equivalents.html#2227" class="Bound">eq</a> <a id="2230" class="Symbol">→</a> <a id="2232" href="Categories.Adjoint.html#5443" class="Function">Radjunct-comm</a> <a id="2246" href="Categories.Adjoint.Equivalents.html#2227" class="Bound">eq</a>
<a id="2257" class="Symbol">}</a>
<a id="2265" class="Symbol">;</a> <a id="2267" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1051" class="Field">iso</a> <a id="2271" class="Symbol">=</a> <a id="2273" class="Symbol">λ</a> <a id="2275" href="Categories.Adjoint.Equivalents.html#2275" class="Bound">_</a> <a id="2277" class="Symbol">→</a> <a id="2279" class="Keyword">record</a>
<a id="2294" class="Symbol">{</a> <a id="2296" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="2301" class="Symbol">=</a> <a id="2303" class="Symbol">λ</a> <a id="2305" href="Categories.Adjoint.Equivalents.html#2305" class="Bound">eq</a> <a id="2308" class="Symbol">→</a> <a id="2310" class="Keyword">let</a> <a id="2314" class="Keyword">open</a> <a id="2319" href="Categories.Category.Core.html#2462" class="Module">D.HomReasoning</a> <a id="2334" class="Keyword">in</a> <a id="2337" href="Categories.Adjoint.html#2072" class="Function">RLadjunct≈id</a> <a id="2350" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2352" href="Categories.Adjoint.Equivalents.html#2305" class="Bound">eq</a>
<a id="2363" class="Symbol">;</a> <a id="2365" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="2370" class="Symbol">=</a> <a id="2372" class="Symbol">λ</a> <a id="2374" href="Categories.Adjoint.Equivalents.html#2374" class="Bound">eq</a> <a id="2377" class="Symbol">→</a> <a id="2379" class="Keyword">let</a> <a id="2383" class="Keyword">open</a> <a id="2388" href="Categories.Category.Core.html#2462" class="Module">C.HomReasoning</a> <a id="2403" class="Keyword">in</a> <a id="2406" href="Categories.Adjoint.html#2579" class="Function">LRadjunct≈id</a> <a id="2419" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2421" href="Categories.Adjoint.Equivalents.html#2374" class="Bound">eq</a>
<a id="2432" class="Symbol">}</a>
<a id="2440" class="Symbol">}</a>
<a id="2445" class="Comment">-- now goes from natural isomorphism back to adjoint.</a>
<a id="2501" class="Comment">-- for simplicity, just construct the case in which homsetoids of C and D</a>
<a id="2577" class="Comment">-- are compatible.</a>
<a id="2599" class="Keyword">private</a>
<a id="2611" href="Categories.Adjoint.Equivalents.html#2611" class="Function">Hom[L-,-]</a> <a id="2621" class="Symbol">:</a> <a id="2623" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="2633" href="Categories.Category.Core.html#3132" class="Function">C.op</a> <a id="2638" href="Categories.Adjoint.Equivalents.html#1619" class="Bound">D</a> <a id="2640" class="Symbol">(</a><a id="2641" href="Categories.Category.Instance.Setoids.html#316" class="Function">Setoids</a> <a id="2649" class="Symbol">_</a> <a id="2651" class="Symbol">_)</a>
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<a id="3238" class="Symbol">(</a><a id="3239" href="Categories.Adjoint.Equivalents.html#2966" class="Function">unitη</a> <a id="3245" href="Categories.Adjoint.Equivalents.html#3215" class="Bound">Y</a> <a id="3247" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="3251" href="Categories.Category.Core.html#630" class="Function">D.id</a><a id="3255" class="Symbol">)</a> <a id="3257" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3259" href="Categories.Adjoint.Equivalents.html#3218" class="Bound">f</a> <a id="3289" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="3292" href="Categories.Morphism.Reasoning.Core.html#3063" class="Function">introˡ</a> <a id="3299" href="Categories.Functor.Core.html#511" class="Field">R.identity</a> <a id="3310" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="3553" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="3557" class="Symbol">(</a><a id="3558" href="Categories.Adjoint.Equivalents.html#3211" class="Bound">X</a> <a id="3560" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="3562" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="3567" href="Categories.Adjoint.Equivalents.html#3215" class="Bound">Y</a><a id="3568" class="Symbol">)</a> <a id="3570" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="3574" href="Categories.Functor.Core.html#455" class="Function">L.F₁</a> <a id="3579" href="Categories.Adjoint.Equivalents.html#3218" class="Bound">f</a> <a id="3604" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="3607" href="Function.Equality.html#1140" class="Field">cong</a> <a id="3612" class="Symbol">(</a><a id="3613" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="3617" class="Symbol">(</a><a id="3618" href="Categories.Adjoint.Equivalents.html#3211" class="Bound">X</a> <a id="3620" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="3622" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="3627" href="Categories.Adjoint.Equivalents.html#3215" class="Bound">Y</a><a id="3628" class="Symbol">))</a> <a id="3631" class="Symbol">(</a><a id="3632" href="Categories.Morphism.Reasoning.Core.html#2899" class="Function">MR.introʳ</a> <a id="3642" href="Categories.Adjoint.Equivalents.html#1619" class="Bound">D</a> <a id="3644" class="Symbol">(</a><a id="3645" href="Categories.Morphism.Reasoning.Core.html#2786" class="Function">MR.elimʳ</a> <a id="3654" href="Categories.Adjoint.Equivalents.html#1619" class="Bound">D</a> <a id="3656" href="Categories.Functor.Core.html#511" class="Function">L.identity</a><a id="3666" class="Symbol">))</a> <a id="3669" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="5319" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5321" href="Categories.Adjoint.Equivalents.html#4129" class="Function">counit</a> <a id="5328" class="Symbol">(</a><a id="5329" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="5334" href="Categories.Adjoint.Equivalents.html#5193" class="Bound">A</a><a id="5335" class="Symbol">)</a> <a id="5337" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5339" href="Categories.Functor.Core.html#455" class="Function">L.F₁</a> <a id="5344" class="Symbol">(</a><a id="5345" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5347" href="Categories.Adjoint.Equivalents.html#3074" class="Function">unit</a> <a id="5352" href="Categories.Adjoint.Equivalents.html#5193" class="Bound">A</a><a id="5353" class="Symbol">)</a> <a id="5360" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5364" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a> <a id="5374" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="5386" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="5389" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5391" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5393" href="Categories.Adjoint.Equivalents.html#4129" class="Function">counit</a> <a id="5400" class="Symbol">(</a><a id="5401" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="5406" href="Categories.Adjoint.Equivalents.html#5193" class="Bound">A</a><a id="5407" class="Symbol">)</a> <a id="5409" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5411" href="Categories.Functor.Core.html#455" class="Function">L.F₁</a> <a id="5416" class="Symbol">(</a><a id="5417" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5419" href="Categories.Adjoint.Equivalents.html#3074" class="Function">unit</a> <a id="5424" href="Categories.Adjoint.Equivalents.html#5193" class="Bound">A</a><a id="5425" class="Symbol">)</a> <a id="5427" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5431" href="Categories.NaturalTransformation.Core.html#827" class="Function">⇐.commute</a> <a id="5441" class="Symbol">(</a><a id="5442" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5444" href="Categories.Adjoint.Equivalents.html#3074" class="Function">unit</a> <a id="5449" href="Categories.Adjoint.Equivalents.html#5193" class="Bound">A</a> <a id="5451" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="5453" href="Categories.Category.Core.html#630" class="Function">id</a><a id="5455" class="Symbol">)</a> <a id="5457" href="Relation.Binary.Structures.html#1577" class="Function">C.Equiv.refl</a> <a id="5470" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="5482" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="5486" class="Symbol">(</a><a id="5487" href="Categories.Adjoint.Equivalents.html#5193" class="Bound">A</a> <a id="5489" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="5491" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="5496" href="Categories.Adjoint.Equivalents.html#5193" class="Bound">A</a><a id="5497" class="Symbol">)</a> <a id="5499" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="5503" class="Symbol">(</a><a id="5504" href="Categories.Functor.Core.html#455" class="Field">R.F₁</a> <a id="5509" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="5512" href="Categories.Category.Core.html#656" class="Function Operator">C.∘</a> <a id="5516" href="Categories.Category.Core.html#630" class="Function">C.id</a> <a id="5521" href="Categories.Category.Core.html#656" class="Function Operator">C.∘</a> <a id="5525" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5527" href="Categories.Adjoint.Equivalents.html#3074" class="Function">unit</a> <a id="5532" href="Categories.Adjoint.Equivalents.html#5193" class="Bound">A</a><a id="5533" class="Symbol">)</a>
<a id="5586" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="5589" href="Function.Equality.html#1140" class="Field">cong</a> <a id="5594" class="Symbol">(</a><a id="5595" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="5599" class="Symbol">(</a><a id="5600" href="Categories.Adjoint.Equivalents.html#5193" class="Bound">A</a> <a id="5602" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="5604" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="5609" href="Categories.Adjoint.Equivalents.html#5193" class="Bound">A</a><a id="5610" class="Symbol">))</a> <a id="5613" class="Symbol">(</a><a id="5614" href="Relation.Binary.Structures.html#1629" class="Function">C.Equiv.trans</a> <a id="5628" class="Symbol">(</a><a id="5629" href="Categories.Morphism.Reasoning.Core.html#2948" class="Function">MR.elimˡ</a> <a id="5638" href="Categories.Adjoint.Equivalents.html#1598" class="Bound">C</a> <a id="5640" href="Categories.Functor.Core.html#511" class="Field">R.identity</a><a id="5650" class="Symbol">)</a> <a id="5652" href="Categories.Category.Core.html#1096" class="Function">C.identityˡ</a><a id="5663" class="Symbol">)</a> <a id="5665" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="5677" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="5681" class="Symbol">(</a><a id="5682" href="Categories.Adjoint.Equivalents.html#5193" class="Bound">A</a> <a id="5684" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="5686" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="5691" href="Categories.Adjoint.Equivalents.html#5193" class="Bound">A</a><a id="5692" class="Symbol">)</a> <a id="5694" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="5698" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5700" href="Categories.Adjoint.Equivalents.html#3074" class="Function">unit</a> <a id="5705" href="Categories.Adjoint.Equivalents.html#5193" class="Bound">A</a> <a id="5718" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="5721" href="Categories.Morphism.html#1586" class="Function">isoˡ</a> <a id="5726" href="Relation.Binary.Structures.html#1577" class="Function">refl</a> <a id="5731" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="5743" href="Categories.Category.Core.html#630" class="Function">id</a>
<a id="5797" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="5805" class="Symbol">;</a> <a id="5807" href="Categories.Adjoint.html#1788" class="Field">zag</a> <a id="5814" class="Symbol">=</a> <a id="5816" class="Symbol">λ</a> <a id="5818" class="Symbol">{</a><a id="5819" href="Categories.Adjoint.Equivalents.html#5819" class="Bound">B</a><a id="5820" class="Symbol">}</a> <a id="5822" class="Symbol">→</a>
<a id="5832" class="Keyword">let</a> <a id="5836" class="Keyword">open</a> <a id="5841" href="Categories.Adjoint.Equivalents.html#1703" class="Module">C</a>
<a id="5855" class="Keyword">open</a> <a id="5860" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="5885" class="Keyword">open</a> <a id="5890" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="5908" class="Keyword">open</a> <a id="5913" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="5916" href="Categories.Adjoint.Equivalents.html#1598" class="Bound">C</a>
<a id="5926" class="Keyword">in</a> <a id="5929" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="5945" href="Categories.Functor.Core.html#455" class="Field">R.F₁</a> <a id="5950" class="Symbol">(</a><a id="5951" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5953" href="Categories.Adjoint.Equivalents.html#4129" class="Function">counit</a> <a id="5960" href="Categories.Adjoint.Equivalents.html#5819" class="Bound">B</a><a id="5961" class="Symbol">)</a> <a id="5963" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5965" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="5967" href="Categories.Adjoint.Equivalents.html#3074" class="Function">unit</a> <a id="5972" class="Symbol">(</a><a id="5973" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="5978" href="Categories.Adjoint.Equivalents.html#5819" class="Bound">B</a><a id="5979" class="Symbol">)</a> <a id="5986" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5990" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="5998" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a> <a id="6008" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="6020" href="Categories.Functor.Core.html#455" class="Field">R.F₁</a> <a id="6025" class="Symbol">(</a><a id="6026" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="6028" href="Categories.Adjoint.Equivalents.html#4129" class="Function">counit</a> <a id="6035" href="Categories.Adjoint.Equivalents.html#5819" class="Bound">B</a><a id="6036" class="Symbol">)</a> <a id="6038" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6040" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="6042" href="Categories.Adjoint.Equivalents.html#3074" class="Function">unit</a> <a id="6047" class="Symbol">(</a><a id="6048" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="6053" href="Categories.Adjoint.Equivalents.html#5819" class="Bound">B</a><a id="6054" class="Symbol">)</a> <a id="6056" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6058" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6061" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="6065" href="Categories.NaturalTransformation.Core.html#827" class="Function">⇒.commute</a> <a id="6075" class="Symbol">(</a><a id="6076" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6079" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="6081" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="6083" href="Categories.Adjoint.Equivalents.html#4129" class="Function">counit</a> <a id="6090" href="Categories.Adjoint.Equivalents.html#5819" class="Bound">B</a><a id="6091" class="Symbol">)</a> <a id="6093" href="Relation.Binary.Structures.html#1577" class="Function">D.Equiv.refl</a> <a id="6106" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="6118" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="6122" class="Symbol">(</a><a id="6123" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="6128" href="Categories.Adjoint.Equivalents.html#5819" class="Bound">B</a> <a id="6130" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="6132" href="Categories.Adjoint.Equivalents.html#5819" class="Bound">B</a><a id="6133" class="Symbol">)</a> <a id="6135" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="6139" class="Symbol">(</a><a id="6140" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="6142" href="Categories.Adjoint.Equivalents.html#4129" class="Function">counit</a> <a id="6149" href="Categories.Adjoint.Equivalents.html#5819" class="Bound">B</a> <a id="6151" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="6155" href="Categories.Category.Core.html#630" class="Function">D.id</a> <a id="6160" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="6164" href="Categories.Functor.Core.html#455" class="Function">L.F₁</a> <a id="6169" href="Categories.Category.Core.html#630" class="Function">id</a><a id="6171" class="Symbol">)</a>
<a id="6224" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6227" href="Function.Equality.html#1140" class="Field">cong</a> <a id="6232" class="Symbol">(</a><a id="6233" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="6237" class="Symbol">(</a><a id="6238" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="6243" href="Categories.Adjoint.Equivalents.html#5819" class="Bound">B</a> <a id="6245" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="6247" href="Categories.Adjoint.Equivalents.html#5819" class="Bound">B</a><a id="6248" class="Symbol">))</a> <a id="6251" class="Symbol">(</a><a id="6252" href="Categories.Morphism.Reasoning.Core.html#2786" class="Function">MR.elimʳ</a> <a id="6261" href="Categories.Adjoint.Equivalents.html#1619" class="Bound">D</a> <a id="6263" class="Symbol">(</a><a id="6264" href="Categories.Morphism.Reasoning.Core.html#2786" class="Function">MR.elimʳ</a> <a id="6273" href="Categories.Adjoint.Equivalents.html#1619" class="Bound">D</a> <a id="6275" href="Categories.Functor.Core.html#511" class="Function">L.identity</a><a id="6285" class="Symbol">))</a> <a id="6288" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="6300" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="6304" class="Symbol">(</a><a id="6305" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="6310" href="Categories.Adjoint.Equivalents.html#5819" class="Bound">B</a> <a id="6312" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="6314" href="Categories.Adjoint.Equivalents.html#5819" class="Bound">B</a><a id="6315" class="Symbol">)</a> <a id="6317" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="6321" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="6323" href="Categories.Adjoint.Equivalents.html#4129" class="Function">counit</a> <a id="6330" href="Categories.Adjoint.Equivalents.html#5819" class="Bound">B</a> <a id="6341" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6344" href="Categories.Morphism.html#1612" class="Function">isoʳ</a> <a id="6349" href="Relation.Binary.Structures.html#1577" class="Function">refl</a> <a id="6354" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="6366" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6407" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="6415" class="Symbol">}</a>
<a id="6423" class="Keyword">where</a> <a id="6429" class="Keyword">module</a> <a id="6436" href="Categories.Adjoint.Equivalents.html#6436" class="Module">i</a> <a id="6438" class="Symbol">{</a><a id="6439" href="Categories.Adjoint.Equivalents.html#6439" class="Bound">X</a><a id="6440" class="Symbol">}</a> <a id="6442" class="Symbol">=</a> <a id="6444" href="Categories.Morphism.html#1528" class="Module">Iso</a> <a id="6448" class="Symbol">(</a><a id="6449" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1051" class="Field">iso</a> <a id="6453" href="Categories.Adjoint.Equivalents.html#6439" class="Bound">X</a><a id="6454" class="Symbol">)</a>
<a id="6468" class="Keyword">open</a> <a id="6473" href="Categories.Adjoint.Equivalents.html#6436" class="Module">i</a>
<a id="6476" class="Comment">-- the general case from isomorphic Hom setoids to adjoint functors</a>
<a id="6544" class="Keyword">module</a> <a id="6551" href="Categories.Adjoint.Equivalents.html#6551" class="Module">_</a> <a id="6553" class="Symbol">{</a><a id="6554" href="Categories.Adjoint.Equivalents.html#6554" class="Bound">C</a> <a id="6556" class="Symbol">:</a> <a id="6558" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="6567" href="Categories.Adjoint.Equivalents.html#1307" class="Generalizable">o</a> <a id="6569" href="Categories.Adjoint.Equivalents.html#1315" class="Generalizable">ℓ</a> <a id="6571" href="Categories.Adjoint.Equivalents.html#1323" class="Generalizable">e</a><a id="6572" class="Symbol">}</a> <a id="6574" class="Symbol">{</a><a id="6575" href="Categories.Adjoint.Equivalents.html#6575" class="Bound">D</a> <a id="6577" class="Symbol">:</a> <a id="6579" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="6588" href="Categories.Adjoint.Equivalents.html#1309" class="Generalizable">o′</a> <a id="6591" href="Categories.Adjoint.Equivalents.html#1317" class="Generalizable">ℓ′</a> <a id="6594" href="Categories.Adjoint.Equivalents.html#1325" class="Generalizable">e′</a><a id="6596" class="Symbol">}</a> <a id="6598" class="Symbol">{</a><a id="6599" href="Categories.Adjoint.Equivalents.html#6599" class="Bound">L</a> <a id="6601" class="Symbol">:</a> <a id="6603" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="6611" href="Categories.Adjoint.Equivalents.html#6554" class="Bound">C</a> <a id="6613" href="Categories.Adjoint.Equivalents.html#6575" class="Bound">D</a><a id="6614" class="Symbol">}</a> <a id="6616" class="Symbol">{</a><a id="6617" href="Categories.Adjoint.Equivalents.html#6617" class="Bound">R</a> <a id="6619" class="Symbol">:</a> <a id="6621" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="6629" href="Categories.Adjoint.Equivalents.html#6575" class="Bound">D</a> <a id="6631" href="Categories.Adjoint.Equivalents.html#6554" class="Bound">C</a><a id="6632" class="Symbol">}</a> <a id="6634" class="Keyword">where</a>
<a id="6642" class="Keyword">private</a>
<a id="6654" class="Keyword">module</a> <a id="6661" href="Categories.Adjoint.Equivalents.html#6661" class="Module">C</a> <a id="6663" class="Symbol">=</a> <a id="6665" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="6674" href="Categories.Adjoint.Equivalents.html#6554" class="Bound">C</a>
<a id="6680" class="Keyword">module</a> <a id="6687" href="Categories.Adjoint.Equivalents.html#6687" class="Module">D</a> <a id="6689" class="Symbol">=</a> <a id="6691" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="6700" href="Categories.Adjoint.Equivalents.html#6575" class="Bound">D</a>
<a id="6706" class="Keyword">module</a> <a id="6713" href="Categories.Adjoint.Equivalents.html#6713" class="Module">L</a> <a id="6715" class="Symbol">=</a> <a id="6717" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="6725" href="Categories.Adjoint.Equivalents.html#6599" class="Bound">L</a>
<a id="6731" class="Keyword">module</a> <a id="6738" href="Categories.Adjoint.Equivalents.html#6738" class="Module">R</a> <a id="6740" class="Symbol">=</a> <a id="6742" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="6750" href="Categories.Adjoint.Equivalents.html#6617" class="Bound">R</a>
<a id="6756" class="Keyword">open</a> <a id="6761" href="Categories.Functor.Core.html#248" class="Module">Functor</a>
<a id="6773" class="Keyword">open</a> <a id="6778" href="Function.Equality.html#898" class="Module">Π</a>
<a id="6785" href="Categories.Adjoint.Equivalents.html#6785" class="Function">Hom[L-,-]</a> <a id="6795" class="Symbol">:</a> <a id="6797" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="6807" href="Categories.Category.Core.html#3132" class="Function">C.op</a> <a id="6812" href="Categories.Adjoint.Equivalents.html#6575" class="Bound">D</a> <a id="6814" class="Symbol">(</a><a id="6815" href="Categories.Category.Instance.Setoids.html#316" class="Function">Setoids</a> <a id="6823" class="Symbol">_</a> <a id="6825" class="Symbol">_)</a>
<a id="6832" href="Categories.Adjoint.Equivalents.html#6785" class="Function">Hom[L-,-]</a> <a id="6842" class="Symbol">=</a> <a id="6844" href="Categories.Functor.Construction.LiftSetoids.html#957" class="Function">LiftSetoids</a> <a id="6856" href="Categories.Adjoint.Equivalents.html#6569" class="Bound">ℓ</a> <a id="6858" href="Categories.Adjoint.Equivalents.html#6571" class="Bound">e</a> <a id="6860" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="6863" href="Categories.Functor.Hom.html#2973" class="Function Operator">Hom[</a> <a id="6868" href="Categories.Adjoint.Equivalents.html#6575" class="Bound">D</a> <a id="6870" href="Categories.Functor.Hom.html#2973" class="Function Operator">][-,-]</a> <a id="6877" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="6880" class="Symbol">(</a><a id="6881" href="Categories.Functor.Core.html#816" class="Function">L.op</a> <a id="6886" href="Categories.Category.Product.html#1962" class="Function Operator">⁂</a> <a id="6888" href="Categories.Adjoint.Equivalents.html#805" class="Function">idF</a><a id="6891" class="Symbol">)</a>
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<a id="9736" class="Keyword">open</a> <a id="9741" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="9766" class="Keyword">open</a> <a id="9771" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="9789" class="Keyword">open</a> <a id="9794" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="9797" href="Categories.Adjoint.Equivalents.html#6575" class="Bound">D</a>
<a id="9807" class="Keyword">in</a> <a id="9810" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="9826" href="Level.html#479" class="Field">lower</a> <a id="9832" class="Symbol">(</a><a id="9833" href="Categories.Adjoint.Equivalents.html#8341" class="Function">counitη</a> <a id="9841" class="Symbol">(</a><a id="9842" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="9847" href="Categories.Adjoint.Equivalents.html#9700" class="Bound">A</a><a id="9848" class="Symbol">)</a> <a id="9850" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="9854" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="9859" href="Categories.Category.Core.html#630" class="Function">C.id</a><a id="9863" class="Symbol">)</a> <a id="9865" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9867" href="Categories.Functor.Core.html#455" class="Function">L.F₁</a> <a id="9872" class="Symbol">(</a><a id="9873" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="9875" href="Categories.Adjoint.Equivalents.html#7243" class="Function">unit</a> <a id="9880" href="Categories.Adjoint.Equivalents.html#9700" class="Bound">A</a><a id="9881" class="Symbol">)</a>
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<a id="10540" href="Categories.Functor.Core.html#455" class="Field">R.F₁</a> <a id="10545" class="Symbol">(</a><a id="10546" href="Level.html#479" class="Field">lower</a> <a id="10552" class="Symbol">(</a><a id="10553" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="10557" class="Symbol">(</a><a id="10558" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="10563" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a> <a id="10565" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="10567" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a><a id="10568" class="Symbol">)</a> <a id="10570" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="10574" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="10579" href="Categories.Category.Core.html#630" class="Function">id</a><a id="10581" class="Symbol">))</a> <a id="10584" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10586" href="Level.html#479" class="Field">lower</a> <a id="10592" class="Symbol">(</a><a id="10593" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="10597" class="Symbol">(</a><a id="10598" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="10603" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a> <a id="10605" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="10607" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="10612" class="Symbol">(</a><a id="10613" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="10618" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a><a id="10619" class="Symbol">))</a> <a id="10622" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="10626" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="10631" href="Categories.Category.Core.html#630" class="Function">D.id</a><a id="10635" class="Symbol">)</a>
<a id="10649" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="10653" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="10661" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a> <a id="10671" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="10683" href="Categories.Functor.Core.html#455" class="Field">R.F₁</a> <a id="10688" class="Symbol">(</a><a id="10689" href="Level.html#479" class="Field">lower</a> <a id="10695" class="Symbol">(</a><a id="10696" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="10700" class="Symbol">(</a><a id="10701" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="10706" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a> <a id="10708" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="10710" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a><a id="10711" class="Symbol">)</a> <a id="10713" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="10717" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="10722" href="Categories.Category.Core.html#630" class="Function">id</a><a id="10724" class="Symbol">))</a> <a id="10727" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10729" href="Level.html#479" class="Field">lower</a> <a id="10735" class="Symbol">(</a><a id="10736" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="10740" class="Symbol">(</a><a id="10741" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="10746" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a> <a id="10748" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="10750" href="Categories.Functor.Core.html#432" class="Function">L.F₀</a> <a id="10755" class="Symbol">(</a><a id="10756" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="10761" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a><a id="10762" class="Symbol">))</a> <a id="10765" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="10769" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="10774" href="Categories.Category.Core.html#630" class="Function">D.id</a><a id="10778" class="Symbol">)</a> <a id="10780" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10782" href="Categories.Category.Core.html#630" class="Function">id</a>
<a id="10797" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="10801" href="Level.html#479" class="Field">lower</a> <a id="10807" class="Symbol">(</a><a id="10808" href="Categories.NaturalTransformation.Core.html#827" class="Function">⇒.commute</a> <a id="10818" class="Symbol">(</a><a id="10819" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="10822" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="10824" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="10826" href="Categories.Adjoint.Equivalents.html#8453" class="Function">counit</a> <a id="10833" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a><a id="10834" class="Symbol">)</a> <a id="10836" class="Symbol">(</a><a id="10837" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="10842" href="Relation.Binary.Structures.html#1577" class="Function">D.Equiv.refl</a><a id="10854" class="Symbol">))</a> <a id="10857" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="10869" href="Level.html#479" class="Field">lower</a> <a id="10875" class="Symbol">(</a><a id="10876" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="10880" class="Symbol">(</a><a id="10881" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="10886" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a> <a id="10888" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="10890" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a><a id="10891" class="Symbol">)</a> <a id="10893" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="10897" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="10902" class="Symbol">(</a><a id="10903" href="Level.html#479" class="Field">lower</a> <a id="10909" class="Symbol">(</a><a id="10910" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="10914" class="Symbol">(</a><a id="10915" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="10920" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a> <a id="10922" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="10924" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a><a id="10925" class="Symbol">)</a> <a id="10927" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="10931" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="10936" href="Categories.Category.Core.html#630" class="Function">id</a><a id="10938" class="Symbol">)</a> <a id="10940" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="10944" href="Categories.Category.Core.html#630" class="Function">D.id</a> <a id="10949" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="10953" href="Categories.Functor.Core.html#455" class="Function">L.F₁</a> <a id="10958" href="Categories.Category.Core.html#630" class="Function">id</a><a id="10960" class="Symbol">))</a>
<a id="10975" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="10978" href="Level.html#479" class="Field">lower</a> <a id="10984" class="Symbol">(</a><a id="10985" href="Function.Equality.html#1140" class="Field">cong</a> <a id="10990" class="Symbol">(</a><a id="10991" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="10995" class="Symbol">(</a><a id="10996" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="11001" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a> <a id="11003" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="11005" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a><a id="11006" class="Symbol">))</a> <a id="11009" class="Symbol">(</a><a id="11010" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="11015" class="Symbol">(</a><a id="11016" href="Categories.Morphism.Reasoning.Core.html#2786" class="Function">MR.elimʳ</a> <a id="11025" href="Categories.Adjoint.Equivalents.html#6575" class="Bound">D</a> <a id="11027" class="Symbol">(</a><a id="11028" href="Categories.Morphism.Reasoning.Core.html#2786" class="Function">MR.elimʳ</a> <a id="11037" href="Categories.Adjoint.Equivalents.html#6575" class="Bound">D</a> <a id="11039" href="Categories.Functor.Core.html#511" class="Function">L.identity</a><a id="11049" class="Symbol">))))</a> <a id="11054" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="11066" href="Level.html#479" class="Field">lower</a> <a id="11072" class="Symbol">(</a><a id="11073" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="11077" class="Symbol">(</a><a id="11078" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="11083" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a> <a id="11085" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="11087" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a><a id="11088" class="Symbol">)</a> <a id="11090" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="11094" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="11099" class="Symbol">(</a><a id="11100" href="Level.html#479" class="Field">lower</a> <a id="11106" class="Symbol">(</a><a id="11107" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="11111" class="Symbol">(</a><a id="11112" href="Categories.Functor.Core.html#432" class="Field">R.F₀</a> <a id="11117" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a> <a id="11119" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="11121" href="Categories.Adjoint.Equivalents.html#10414" class="Bound">B</a><a id="11122" class="Symbol">)</a> <a id="11124" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="11128" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="11133" href="Categories.Category.Core.html#630" class="Function">id</a><a id="11135" class="Symbol">)))</a>
<a id="11151" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="11154" href="Level.html#479" class="Field">lower</a> <a id="11160" class="Symbol">(</a><a id="11161" href="Categories.Morphism.html#1612" class="Function">isoʳ</a> <a id="11166" class="Symbol">(</a><a id="11167" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="11172" href="Relation.Binary.Structures.html#1577" class="Function">refl</a><a id="11176" class="Symbol">))</a> <a id="11179" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="11191" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="11194" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="11202" class="Symbol">}</a>
<a id="11210" class="Keyword">where</a> <a id="11216" class="Keyword">open</a> <a id="11221" href="Categories.NaturalTransformation.Core.html#466" class="Module">NaturalTransformation</a>
<a id="11255" class="Keyword">module</a> <a id="11262" href="Categories.Adjoint.Equivalents.html#11262" class="Module">_</a> <a id="11264" class="Symbol">{</a><a id="11265" href="Categories.Adjoint.Equivalents.html#11265" class="Bound">X</a><a id="11266" class="Symbol">}</a> <a id="11268" class="Keyword">where</a>
<a id="11288" class="Keyword">open</a> <a id="11293" href="Categories.Morphism.html#1528" class="Module">Iso</a> <a id="11297" class="Symbol">(</a><a id="11298" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1051" class="Field">iso</a> <a id="11302" href="Categories.Adjoint.Equivalents.html#11265" class="Bound">X</a><a id="11303" class="Symbol">)</a> <a id="11305" class="Keyword">public</a>
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