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<html><head><meta charset="utf-8"><title>Categories.Category.Monoidal.Closed</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Symbol">{-#</a> <a id="5" class="Keyword">OPTIONS</a> <a id="13" class="Pragma">--without-K</a> <a id="25" class="Pragma">--safe</a> <a id="32" class="Symbol">#-}</a>
<a id="37" class="Keyword">open</a> <a id="42" class="Keyword">import</a> <a id="49" href="Categories.Category.html" class="Module">Categories.Category</a>
<a id="69" class="Keyword">open</a> <a id="74" class="Keyword">import</a> <a id="81" href="Categories.Category.Monoidal.html" class="Module">Categories.Category.Monoidal</a>
<a id="111" class="Comment">-- the definition used here is not very similar to what one usually sees in nLab or</a>
<a id="195" class="Comment">-- any textbook. the difference is that usually closed monoidal category is defined</a>
<a id="280" class="Comment">-- through a right adjoint of -⊗X, which is [X,-]. then there is an induced bifunctor</a>
<a id="366" class="Comment">-- [-,-].</a>
<a id="376" class="Comment">--</a>
<a id="379" class="Comment">-- but in proof relevant context, the induced bifunctor [-,-] does not have to be</a>
<a id="461" class="Comment">-- exactly the intended bifunctor! in fact, one can probably only show that the</a>
<a id="541" class="Comment">-- intended bifunctor is only naturally isomorphic to the induced one, which is</a>
<a id="621" class="Comment">-- significantly weaker.</a>
<a id="646" class="Comment">--</a>
<a id="649" class="Comment">-- the approach taken here as an alternative is to BEGIN with a bifunctor</a>
<a id="723" class="Comment">-- already. however, is it required to have mates between any given two adjoints. this</a>
<a id="810" class="Comment">-- definition can be shown equivalent to the previous one but just works better.</a>
<a id="891" class="Keyword">module</a> <a id="898" href="Categories.Category.Monoidal.Closed.html" class="Module">Categories.Category.Monoidal.Closed</a> <a id="934" class="Symbol">{</a><a id="935" href="Categories.Category.Monoidal.Closed.html#935" class="Bound">o</a> <a id="937" href="Categories.Category.Monoidal.Closed.html#937" class="Bound">ℓ</a> <a id="939" href="Categories.Category.Monoidal.Closed.html#939" class="Bound">e</a><a id="940" class="Symbol">}</a> <a id="942" class="Symbol">{</a><a id="943" href="Categories.Category.Monoidal.Closed.html#943" class="Bound">C</a> <a id="945" class="Symbol">:</a> <a id="947" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="956" href="Categories.Category.Monoidal.Closed.html#935" class="Bound">o</a> <a id="958" href="Categories.Category.Monoidal.Closed.html#937" class="Bound">ℓ</a> <a id="960" href="Categories.Category.Monoidal.Closed.html#939" class="Bound">e</a><a id="961" class="Symbol">}</a> <a id="963" class="Symbol">(</a><a id="964" href="Categories.Category.Monoidal.Closed.html#964" class="Bound">M</a> <a id="966" class="Symbol">:</a> <a id="968" href="Categories.Category.Monoidal.Core.html#1287" class="Record">Monoidal</a> <a id="977" href="Categories.Category.Monoidal.Closed.html#943" class="Bound">C</a><a id="978" class="Symbol">)</a> <a id="980" class="Keyword">where</a>
<a id="987" class="Keyword">private</a>
<a id="997" class="Keyword">module</a> <a id="C"></a><a id="1004" href="Categories.Category.Monoidal.Closed.html#1004" class="Module">C</a> <a id="1006" class="Symbol">=</a> <a id="1008" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1017" href="Categories.Category.Monoidal.Closed.html#943" class="Bound">C</a>
<a id="1021" class="Keyword">open</a> <a id="1026" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1035" href="Categories.Category.Monoidal.Closed.html#943" class="Bound">C</a>
<a id="1040" class="Keyword">variable</a>
<a id="1053" href="Categories.Category.Monoidal.Closed.html#1053" class="Generalizable">X</a> <a id="1055" href="Categories.Category.Monoidal.Closed.html#1055" class="Generalizable">Y</a> <a id="1057" href="Categories.Category.Monoidal.Closed.html#1057" class="Generalizable">A</a> <a id="1059" href="Categories.Category.Monoidal.Closed.html#1059" class="Generalizable">B</a> <a id="1061" class="Symbol">:</a> <a id="1063" href="Categories.Category.Core.html#559" class="Function">Obj</a>
<a id="1068" class="Keyword">open</a> <a id="1073" class="Keyword">import</a> <a id="1080" href="Level.html" class="Module">Level</a>
<a id="1086" class="Keyword">open</a> <a id="1091" class="Keyword">import</a> <a id="1098" href="Data.Product.html" class="Module">Data.Product</a> <a id="1111" class="Keyword">using</a> <a id="1117" class="Symbol">(</a><a id="1118" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="1121" class="Symbol">)</a>
<a id="1124" class="Keyword">open</a> <a id="1129" class="Keyword">import</a> <a id="1136" href="Categories.Adjoint.html" class="Module">Categories.Adjoint</a>
<a id="1155" class="Keyword">open</a> <a id="1160" class="Keyword">import</a> <a id="1167" href="Categories.Adjoint.Equivalents.html" class="Module">Categories.Adjoint.Equivalents</a> <a id="1198" class="Keyword">using</a> <a id="1204" class="Symbol">(</a><a id="1205" href="Categories.Adjoint.Equivalents.html#1961" class="Function">Hom-NI′</a><a id="1212" class="Symbol">)</a>
<a id="1214" class="Keyword">open</a> <a id="1219" class="Keyword">import</a> <a id="1226" href="Categories.Adjoint.Mate.html" class="Module">Categories.Adjoint.Mate</a>
<a id="1250" class="Keyword">open</a> <a id="1255" class="Keyword">import</a> <a id="1262" href="Categories.Functor.html" class="Module">Categories.Functor</a> <a id="1281" class="Keyword">renaming</a> <a id="1290" class="Symbol">(</a><a id="1291" href="Categories.Functor.html#349" class="Function">id</a> <a id="1294" class="Symbol">to</a> <a id="1297" class="Function">idF</a><a id="1300" class="Symbol">)</a>
<a id="1302" class="Keyword">open</a> <a id="1307" class="Keyword">import</a> <a id="1314" href="Categories.Functor.Bifunctor.html" class="Module">Categories.Functor.Bifunctor</a>
<a id="1343" class="Keyword">open</a> <a id="1348" class="Keyword">import</a> <a id="1355" href="Categories.Functor.Hom.html" class="Module">Categories.Functor.Hom</a>
<a id="1378" class="Keyword">open</a> <a id="1383" class="Keyword">import</a> <a id="1390" href="Categories.Category.Instance.Setoids.html" class="Module">Categories.Category.Instance.Setoids</a>
<a id="1427" class="Keyword">open</a> <a id="1432" class="Keyword">import</a> <a id="1439" href="Categories.NaturalTransformation.html" class="Module">Categories.NaturalTransformation</a> <a id="1472" class="Keyword">hiding</a> <a id="1479" class="Symbol">(</a><a id="1480" href="Categories.NaturalTransformation.Core.html#2132" class="Function">id</a><a id="1482" class="Symbol">)</a>
<a id="1484" class="Keyword">open</a> <a id="1489" class="Keyword">import</a> <a id="1496" href="Categories.NaturalTransformation.Properties.html" class="Module">Categories.NaturalTransformation.Properties</a>
<a id="1540" class="Keyword">open</a> <a id="1545" class="Keyword">import</a> <a id="1552" href="Categories.NaturalTransformation.NaturalIsomorphism.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism</a> <a id="1604" class="Symbol">as</a> <a id="1607" class="Module">NI</a>
<a id="1611" class="Keyword">record</a> <a id="Closed"></a><a id="1618" href="Categories.Category.Monoidal.Closed.html#1618" class="Record">Closed</a> <a id="1625" class="Symbol">:</a> <a id="1627" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1631" class="Symbol">(</a><a id="1632" href="Level.html#602" class="Function">levelOfTerm</a> <a id="1644" href="Categories.Category.Monoidal.Closed.html#964" class="Bound">M</a><a id="1645" class="Symbol">)</a> <a id="1647" class="Keyword">where</a>
<a id="1655" class="Keyword">open</a> <a id="1660" href="Categories.Category.Monoidal.Core.html#1287" class="Module">Monoidal</a> <a id="1669" href="Categories.Category.Monoidal.Closed.html#964" class="Bound">M</a> <a id="1671" class="Keyword">public</a>
<a id="1681" class="Keyword">field</a>
<a id="Closed.[-,-]"></a><a id="1691" href="Categories.Category.Monoidal.Closed.html#1691" class="Field">[-,-]</a> <a id="1699" class="Symbol">:</a> <a id="1701" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="1711" href="Categories.Category.Core.html#3132" class="Function">C.op</a> <a id="1716" href="Categories.Category.Monoidal.Closed.html#943" class="Bound">C</a> <a id="1718" href="Categories.Category.Monoidal.Closed.html#943" class="Bound">C</a>
<a id="Closed.adjoint"></a><a id="1724" href="Categories.Category.Monoidal.Closed.html#1724" class="Field">adjoint</a> <a id="1732" class="Symbol">:</a> <a id="1734" class="Symbol">(</a><a id="1735" href="Categories.Category.Monoidal.Core.html#1645" class="Function Operator">-⊗</a> <a id="1738" href="Categories.Category.Monoidal.Closed.html#1053" class="Generalizable">X</a><a id="1739" class="Symbol">)</a> <a id="1741" href="Categories.Adjoint.html#7818" class="Function Operator">⊣</a> <a id="1743" href="Categories.Functor.Bifunctor.html#874" class="Function">appˡ</a> <a id="1748" href="Categories.Category.Monoidal.Closed.html#1691" class="Field">[-,-]</a> <a id="1754" href="Categories.Category.Monoidal.Closed.html#1053" class="Generalizable">X</a>
<a id="Closed.mate"></a><a id="1760" href="Categories.Category.Monoidal.Closed.html#1760" class="Field">mate</a> <a id="1768" class="Symbol">:</a> <a id="1770" class="Symbol">(</a><a id="1771" href="Categories.Category.Monoidal.Closed.html#1771" class="Bound">f</a> <a id="1773" class="Symbol">:</a> <a id="1775" href="Categories.Category.Monoidal.Closed.html#1053" class="Generalizable">X</a> <a id="1777" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1779" href="Categories.Category.Monoidal.Closed.html#1055" class="Generalizable">Y</a><a id="1780" class="Symbol">)</a> <a id="1782" class="Symbol">→</a> <a id="1784" href="Categories.Adjoint.Mate.html#1140" class="Record">Mate</a> <a id="1789" class="Symbol">(</a><a id="1790" href="Categories.Category.Monoidal.Closed.html#1724" class="Field">adjoint</a> <a id="1798" class="Symbol">{</a><a id="1799" href="Categories.Category.Monoidal.Closed.html#1053" class="Generalizable">X</a><a id="1800" class="Symbol">})</a> <a id="1803" class="Symbol">(</a><a id="1804" href="Categories.Category.Monoidal.Closed.html#1724" class="Field">adjoint</a> <a id="1812" class="Symbol">{</a><a id="1813" href="Categories.Category.Monoidal.Closed.html#1055" class="Generalizable">Y</a><a id="1814" class="Symbol">})</a> <a id="1817" class="Symbol">(</a><a id="1818" href="Categories.NaturalTransformation.Properties.html#2705" class="Function">appʳ-nat</a> <a id="1827" href="Categories.Category.Monoidal.Core.html#1355" class="Function">⊗</a> <a id="1829" href="Categories.Category.Monoidal.Closed.html#1771" class="Bound">f</a><a id="1830" class="Symbol">)</a> <a id="1832" class="Symbol">(</a><a id="1833" href="Categories.NaturalTransformation.Properties.html#2570" class="Function">appˡ-nat</a> <a id="1842" href="Categories.Category.Monoidal.Closed.html#1691" class="Field">[-,-]</a> <a id="1848" href="Categories.Category.Monoidal.Closed.html#1771" class="Bound">f</a><a id="1849" class="Symbol">)</a>
<a id="1854" class="Keyword">module</a> <a id="Closed.[-,-]"></a><a id="1861" href="Categories.Category.Monoidal.Closed.html#1861" class="Module">[-,-]</a> <a id="1874" class="Symbol">=</a> <a id="1876" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1884" href="Categories.Category.Monoidal.Closed.html#1691" class="Field">[-,-]</a>
<a id="1892" class="Keyword">module</a> <a id="Closed.adjoint"></a><a id="1899" href="Categories.Category.Monoidal.Closed.html#1899" class="Module">adjoint</a> <a id="1907" class="Symbol">{</a><a id="1908" href="Categories.Category.Monoidal.Closed.html#1908" class="Bound">X</a><a id="1909" class="Symbol">}</a> <a id="1912" class="Symbol">=</a> <a id="1914" href="Categories.Adjoint.html#1260" class="Module">Adjoint</a> <a id="1922" class="Symbol">(</a><a id="1923" href="Categories.Category.Monoidal.Closed.html#1724" class="Field">adjoint</a> <a id="1931" class="Symbol">{</a><a id="1932" href="Categories.Category.Monoidal.Closed.html#1908" class="Bound">X</a><a id="1933" class="Symbol">})</a>
<a id="1938" class="Keyword">module</a> <a id="Closed.mate"></a><a id="1945" href="Categories.Category.Monoidal.Closed.html#1945" class="Module">mate</a> <a id="1950" class="Symbol">{</a><a id="1951" href="Categories.Category.Monoidal.Closed.html#1951" class="Bound">X</a> <a id="1953" href="Categories.Category.Monoidal.Closed.html#1953" class="Bound">Y</a><a id="1954" class="Symbol">}</a> <a id="1956" href="Categories.Category.Monoidal.Closed.html#1956" class="Bound">f</a> <a id="1958" class="Symbol">=</a> <a id="1960" href="Categories.Adjoint.Mate.html#1140" class="Module">Mate</a> <a id="1965" class="Symbol">(</a><a id="1966" href="Categories.Category.Monoidal.Closed.html#1760" class="Field">mate</a> <a id="1971" class="Symbol">{</a><a id="1972" href="Categories.Category.Monoidal.Closed.html#1951" class="Bound">X</a><a id="1973" class="Symbol">}</a> <a id="1975" class="Symbol">{</a><a id="1976" href="Categories.Category.Monoidal.Closed.html#1953" class="Bound">Y</a><a id="1977" class="Symbol">}</a> <a id="1979" href="Categories.Category.Monoidal.Closed.html#1956" class="Bound">f</a><a id="1980" class="Symbol">)</a>
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<a id="2013" href="Categories.Category.Monoidal.Closed.html#1985" class="Function Operator">[_,-]</a> <a id="2019" class="Symbol">=</a> <a id="2021" href="Categories.Functor.Bifunctor.html#874" class="Function">appˡ</a> <a id="2026" href="Categories.Category.Monoidal.Closed.html#1691" class="Field">[-,-]</a>
<a id="Closed.[-,_]"></a><a id="2035" href="Categories.Category.Monoidal.Closed.html#2035" class="Function Operator">[-,_]</a> <a id="2041" class="Symbol">:</a> <a id="2043" href="Categories.Category.Core.html#559" class="Function">Obj</a> <a id="2047" class="Symbol">→</a> <a id="2049" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="2057" href="Categories.Category.Core.html#3132" class="Function">C.op</a> <a id="2062" href="Categories.Category.Monoidal.Closed.html#943" class="Bound">C</a>
<a id="2066" href="Categories.Category.Monoidal.Closed.html#2035" class="Function Operator">[-,_]</a> <a id="2072" class="Symbol">=</a> <a id="2074" href="Categories.Functor.Bifunctor.html#938" class="Function">appʳ</a> <a id="2079" href="Categories.Category.Monoidal.Closed.html#1691" class="Field">[-,-]</a>
<a id="Closed.[_,_]₀"></a><a id="2088" href="Categories.Category.Monoidal.Closed.html#2088" class="Function Operator">[_,_]₀</a> <a id="2095" class="Symbol">:</a> <a id="2097" href="Categories.Category.Core.html#559" class="Function">Obj</a> <a id="2101" class="Symbol">→</a> <a id="2103" href="Categories.Category.Core.html#559" class="Function">Obj</a> <a id="2107" class="Symbol">→</a> <a id="2109" href="Categories.Category.Core.html#559" class="Function">Obj</a>
<a id="2115" href="Categories.Category.Monoidal.Closed.html#2088" class="Function Operator">[</a> <a id="2117" href="Categories.Category.Monoidal.Closed.html#2117" class="Bound">X</a> <a id="2119" href="Categories.Category.Monoidal.Closed.html#2088" class="Function Operator">,</a> <a id="2121" href="Categories.Category.Monoidal.Closed.html#2121" class="Bound">Y</a> <a id="2123" href="Categories.Category.Monoidal.Closed.html#2088" class="Function Operator">]₀</a> <a id="2126" class="Symbol">=</a> <a id="2128" href="Categories.Functor.Core.html#432" class="Function">[-,-].F₀</a> <a id="2137" class="Symbol">(</a><a id="2138" href="Categories.Category.Monoidal.Closed.html#2117" class="Bound">X</a> <a id="2140" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2142" href="Categories.Category.Monoidal.Closed.html#2121" class="Bound">Y</a><a id="2143" class="Symbol">)</a>
<a id="Closed.[_,_]₁"></a><a id="2148" href="Categories.Category.Monoidal.Closed.html#2148" class="Function Operator">[_,_]₁</a> <a id="2155" class="Symbol">:</a> <a id="2157" href="Categories.Category.Monoidal.Closed.html#1057" class="Generalizable">A</a> <a id="2159" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2161" href="Categories.Category.Monoidal.Closed.html#1059" class="Generalizable">B</a> <a id="2163" class="Symbol">→</a> <a id="2165" href="Categories.Category.Monoidal.Closed.html#1053" class="Generalizable">X</a> <a id="2167" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2169" href="Categories.Category.Monoidal.Closed.html#1055" class="Generalizable">Y</a> <a id="2171" class="Symbol">→</a> <a id="2173" href="Categories.Category.Monoidal.Closed.html#2088" class="Function Operator">[</a> <a id="2175" href="Categories.Category.Monoidal.Closed.html#1059" class="Generalizable">B</a> <a id="2177" href="Categories.Category.Monoidal.Closed.html#2088" class="Function Operator">,</a> <a id="2179" href="Categories.Category.Monoidal.Closed.html#1053" class="Generalizable">X</a> <a id="2181" href="Categories.Category.Monoidal.Closed.html#2088" class="Function Operator">]₀</a> <a id="2184" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2186" href="Categories.Category.Monoidal.Closed.html#2088" class="Function Operator">[</a> <a id="2188" href="Categories.Category.Monoidal.Closed.html#1057" class="Generalizable">A</a> <a id="2190" href="Categories.Category.Monoidal.Closed.html#2088" class="Function Operator">,</a> <a id="2192" href="Categories.Category.Monoidal.Closed.html#1055" class="Generalizable">Y</a> <a id="2194" href="Categories.Category.Monoidal.Closed.html#2088" class="Function Operator">]₀</a>
<a id="2199" href="Categories.Category.Monoidal.Closed.html#2148" class="Function Operator">[</a> <a id="2201" href="Categories.Category.Monoidal.Closed.html#2201" class="Bound">f</a> <a id="2203" href="Categories.Category.Monoidal.Closed.html#2148" class="Function Operator">,</a> <a id="2205" href="Categories.Category.Monoidal.Closed.html#2205" class="Bound">g</a> <a id="2207" href="Categories.Category.Monoidal.Closed.html#2148" class="Function Operator">]₁</a> <a id="2210" class="Symbol">=</a> <a id="2212" href="Categories.Functor.Core.html#455" class="Function">[-,-].F₁</a> <a id="2221" class="Symbol">(</a><a id="2222" href="Categories.Category.Monoidal.Closed.html#2201" class="Bound">f</a> <a id="2224" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2226" href="Categories.Category.Monoidal.Closed.html#2205" class="Bound">g</a><a id="2227" class="Symbol">)</a>
<a id="Closed.Hom[-⊗_,-]"></a><a id="2232" href="Categories.Category.Monoidal.Closed.html#2232" class="Function Operator">Hom[-⊗_,-]</a> <a id="2243" class="Symbol">:</a> <a id="2245" class="Symbol">∀</a> <a id="2247" href="Categories.Category.Monoidal.Closed.html#2247" class="Bound">X</a> <a id="2249" class="Symbol">→</a> <a id="2251" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="2261" href="Categories.Category.Core.html#3132" class="Function">C.op</a> <a id="2266" href="Categories.Category.Monoidal.Closed.html#943" class="Bound">C</a> <a id="2268" class="Symbol">(</a><a id="2269" href="Categories.Category.Instance.Setoids.html#555" class="Function">Setoids</a> <a id="2277" href="Categories.Category.Monoidal.Closed.html#937" class="Bound">ℓ</a> <a id="2279" href="Categories.Category.Monoidal.Closed.html#939" class="Bound">e</a><a id="2280" class="Symbol">)</a>
<a id="2284" href="Categories.Category.Monoidal.Closed.html#2232" class="Function Operator">Hom[-⊗</a> <a id="2291" href="Categories.Category.Monoidal.Closed.html#2291" class="Bound">X</a> <a id="2293" href="Categories.Category.Monoidal.Closed.html#2232" class="Function Operator">,-]</a> <a id="2297" class="Symbol">=</a> <a id="2299" href="Categories.Adjoint.html#3049" class="Function">adjoint.Hom[L-,-]</a> <a id="2317" class="Symbol">{</a><a id="2318" href="Categories.Category.Monoidal.Closed.html#2291" class="Bound">X</a><a id="2319" class="Symbol">}</a>
<a id="Closed.Hom[-,[_,-]]"></a><a id="2324" href="Categories.Category.Monoidal.Closed.html#2324" class="Function Operator">Hom[-,[_,-]]</a> <a id="2337" class="Symbol">:</a> <a id="2339" class="Symbol">∀</a> <a id="2341" href="Categories.Category.Monoidal.Closed.html#2341" class="Bound">X</a> <a id="2343" class="Symbol">→</a> <a id="2345" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="2355" href="Categories.Category.Core.html#3132" class="Function">C.op</a> <a id="2360" href="Categories.Category.Monoidal.Closed.html#943" class="Bound">C</a> <a id="2362" class="Symbol">(</a><a id="2363" href="Categories.Category.Instance.Setoids.html#555" class="Function">Setoids</a> <a id="2371" href="Categories.Category.Monoidal.Closed.html#937" class="Bound">ℓ</a> <a id="2373" href="Categories.Category.Monoidal.Closed.html#939" class="Bound">e</a><a id="2374" class="Symbol">)</a>
<a id="2378" href="Categories.Category.Monoidal.Closed.html#2324" class="Function Operator">Hom[-,[</a> <a id="2386" href="Categories.Category.Monoidal.Closed.html#2386" class="Bound">X</a> <a id="2388" href="Categories.Category.Monoidal.Closed.html#2324" class="Function Operator">,-]]</a> <a id="2393" class="Symbol">=</a> <a id="2395" href="Categories.Adjoint.html#3139" class="Function">adjoint.Hom[-,R-]</a> <a id="2413" class="Symbol">{</a><a id="2414" href="Categories.Category.Monoidal.Closed.html#2386" class="Bound">X</a><a id="2415" class="Symbol">}</a>
<a id="Closed.Hom-NI"></a><a id="2420" href="Categories.Category.Monoidal.Closed.html#2420" class="Function">Hom-NI</a> <a id="2427" class="Symbol">:</a> <a id="2429" class="Symbol">∀</a> <a id="2431" class="Symbol">{</a><a id="2432" href="Categories.Category.Monoidal.Closed.html#2432" class="Bound">X</a> <a id="2434" class="Symbol">:</a> <a id="2436" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="2439" class="Symbol">}</a> <a id="2441" class="Symbol">→</a> <a id="2443" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a> <a id="2462" href="Categories.Category.Monoidal.Closed.html#2232" class="Function Operator">Hom[-⊗</a> <a id="2469" href="Categories.Category.Monoidal.Closed.html#2432" class="Bound">X</a> <a id="2471" href="Categories.Category.Monoidal.Closed.html#2232" class="Function Operator">,-]</a> <a id="2475" href="Categories.Category.Monoidal.Closed.html#2324" class="Function Operator">Hom[-,[</a> <a id="2483" href="Categories.Category.Monoidal.Closed.html#2432" class="Bound">X</a> <a id="2485" href="Categories.Category.Monoidal.Closed.html#2324" class="Function Operator">,-]]</a>
<a id="2492" href="Categories.Category.Monoidal.Closed.html#2420" class="Function">Hom-NI</a> <a id="2499" class="Symbol">=</a> <a id="2501" href="Categories.Adjoint.Equivalents.html#1961" class="Function">Hom-NI′</a> <a id="2509" href="Categories.Category.Monoidal.Closed.html#1724" class="Field">adjoint</a>
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