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<html><head><meta charset="utf-8"><title>Categories.NaturalTransformation.Properties</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Symbol">{-#</a> <a id="5" class="Keyword">OPTIONS</a> <a id="13" class="Pragma">--without-K</a> <a id="25" class="Pragma">--safe</a> <a id="32" class="Symbol">#-}</a>
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<a id="1849" href="Categories.NaturalTransformation.Core.html#783" class="Function">η</a> <a id="1851" href="Categories.NaturalTransformation.Properties.html#1775" class="Bound">Y</a> <a id="1853" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1855" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1858" href="Categories.NaturalTransformation.Properties.html#817" class="Bound">F</a> <a id="1860" href="Categories.NaturalTransformation.Properties.html#1778" class="Bound">f</a> <a id="1862" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1864" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="1868" href="Categories.NaturalTransformation.Properties.html#1773" class="Bound">X</a> <a id="1873" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1876" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="1882" class="Symbol">(</a><a id="1883" href="Categories.NaturalTransformation.Core.html#827" class="Function">commute</a> <a id="1891" href="Categories.NaturalTransformation.Properties.html#1778" class="Bound">f</a><a id="1892" class="Symbol">)</a> <a id="1894" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1896" href="Categories.Category.Core.html#715" class="Function">assoc</a> <a id="1902" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1910" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1913" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">G</a> <a id="1915" href="Categories.NaturalTransformation.Properties.html#1778" class="Bound">f</a> <a id="1917" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1919" href="Categories.NaturalTransformation.Core.html#783" class="Function">η</a> <a id="1921" href="Categories.NaturalTransformation.Properties.html#1773" class="Bound">X</a> <a id="1923" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1925" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="1929" href="Categories.NaturalTransformation.Properties.html#1773" class="Bound">X</a> <a id="1934" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="1940" class="Symbol">}</a>
<a id="1946" class="Keyword">where</a> <a id="1952" class="Keyword">open</a> <a id="1957" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Module">NaturalIsomorphism</a> <a id="1976" href="Categories.NaturalTransformation.Properties.html#1697" class="Bound">F≃F′</a>
<a id="1991" class="Keyword">open</a> <a id="1996" href="Categories.NaturalTransformation.Core.html#466" class="Module">NaturalTransformation</a> <a id="2018" href="Categories.NaturalTransformation.Properties.html#1695" class="Bound">α</a>
<a id="2023" href="Categories.NaturalTransformation.Properties.html#2023" class="Function">replaceʳ</a> <a id="2032" class="Symbol">:</a> <a id="2034" class="Symbol">∀</a> <a id="2036" class="Symbol">{</a><a id="2037" href="Categories.NaturalTransformation.Properties.html#2037" class="Bound">G′</a><a id="2039" class="Symbol">}</a> <a id="2041" class="Symbol">→</a> <a id="2043" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="2065" href="Categories.NaturalTransformation.Properties.html#817" class="Bound">F</a> <a id="2067" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">G</a> <a id="2069" class="Symbol">→</a> <a id="2071" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">G</a> <a id="2073" href="Categories.NaturalTransformation.NaturalIsomorphism.html#3600" class="Function Operator">≃</a> <a id="2075" href="Categories.NaturalTransformation.Properties.html#2037" class="Bound">G′</a> <a id="2078" class="Symbol">→</a> <a id="2080" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="2102" href="Categories.NaturalTransformation.Properties.html#817" class="Bound">F</a> <a id="2104" href="Categories.NaturalTransformation.Properties.html#2037" class="Bound">G′</a>
<a id="2109" href="Categories.NaturalTransformation.Properties.html#2023" class="Function">replaceʳ</a> <a id="2118" class="Symbol">{</a><a id="2119" href="Categories.NaturalTransformation.Properties.html#2119" class="Bound">G′</a><a id="2121" class="Symbol">}</a> <a id="2123" href="Categories.NaturalTransformation.Properties.html#2123" class="Bound">α</a> <a id="2125" href="Categories.NaturalTransformation.Properties.html#2125" class="Bound">G≃G′</a> <a id="2130" class="Symbol">=</a> <a id="2132" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="2141" class="Keyword">record</a>
<a id="2152" class="Symbol">{</a> <a id="2154" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="2162" class="Symbol">=</a> <a id="2164" class="Symbol">λ</a> <a id="2166" href="Categories.NaturalTransformation.Properties.html#2166" class="Bound">X</a> <a id="2168" class="Symbol">→</a> <a id="2170" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="2174" href="Categories.NaturalTransformation.Properties.html#2166" class="Bound">X</a> <a id="2176" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2178" href="Categories.NaturalTransformation.Core.html#783" class="Function">η</a> <a id="2180" href="Categories.NaturalTransformation.Properties.html#2166" class="Bound">X</a>
<a id="2186" class="Symbol">;</a> <a id="2188" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="2196" class="Symbol">=</a> <a id="2198" class="Symbol">λ</a> <a id="2200" class="Symbol">{</a><a id="2201" href="Categories.NaturalTransformation.Properties.html#2201" class="Bound">X</a> <a id="2203" href="Categories.NaturalTransformation.Properties.html#2203" class="Bound">Y</a><a id="2204" class="Symbol">}</a> <a id="2206" href="Categories.NaturalTransformation.Properties.html#2206" class="Bound">f</a> <a id="2208" class="Symbol">→</a> <a id="2210" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="2222" class="Symbol">(</a><a id="2223" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="2227" href="Categories.NaturalTransformation.Properties.html#2203" class="Bound">Y</a> <a id="2229" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2231" href="Categories.NaturalTransformation.Core.html#783" class="Function">η</a> <a id="2233" href="Categories.NaturalTransformation.Properties.html#2203" class="Bound">Y</a><a id="2234" class="Symbol">)</a> <a id="2236" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2238" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="2241" href="Categories.NaturalTransformation.Properties.html#817" class="Bound">F</a> <a id="2243" href="Categories.NaturalTransformation.Properties.html#2206" class="Bound">f</a> <a id="2245" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2248" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="2254" class="Symbol">(</a><a id="2255" href="Categories.NaturalTransformation.Core.html#827" class="Function">commute</a> <a id="2263" href="Categories.NaturalTransformation.Properties.html#2206" class="Bound">f</a><a id="2264" class="Symbol">)</a> <a id="2266" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2274" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="2278" href="Categories.NaturalTransformation.Properties.html#2203" class="Bound">Y</a> <a id="2280" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2282" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="2285" href="Categories.NaturalTransformation.Properties.html#819" class="Bound">G</a> <a id="2287" href="Categories.NaturalTransformation.Properties.html#2206" class="Bound">f</a> <a id="2289" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2291" href="Categories.NaturalTransformation.Core.html#783" class="Function">η</a> <a id="2293" href="Categories.NaturalTransformation.Properties.html#2201" class="Bound">X</a> <a id="2297" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2300" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="2306" class="Symbol">(</a><a id="2307" href="Categories.NaturalTransformation.Core.html#827" class="Function">⇒.commute</a> <a id="2317" href="Categories.NaturalTransformation.Properties.html#2206" class="Bound">f</a><a id="2318" class="Symbol">)</a> <a id="2320" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2322" href="Categories.Category.Core.html#715" class="Function">assoc</a> <a id="2328" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2336" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="2339" href="Categories.NaturalTransformation.Properties.html#2119" class="Bound">G′</a> <a id="2342" href="Categories.NaturalTransformation.Properties.html#2206" class="Bound">f</a> <a id="2344" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2346" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="2350" href="Categories.NaturalTransformation.Properties.html#2201" class="Bound">X</a> <a id="2352" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2354" href="Categories.NaturalTransformation.Core.html#783" class="Function">η</a> <a id="2356" href="Categories.NaturalTransformation.Properties.html#2201" class="Bound">X</a> <a id="2359" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="2365" class="Symbol">}</a>
<a id="2371" class="Keyword">where</a> <a id="2377" class="Keyword">open</a> <a id="2382" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Module">NaturalIsomorphism</a> <a id="2401" href="Categories.NaturalTransformation.Properties.html#2125" class="Bound">G≃G′</a>
<a id="2416" class="Keyword">open</a> <a id="2421" href="Categories.NaturalTransformation.Core.html#466" class="Module">NaturalTransformation</a> <a id="2443" href="Categories.NaturalTransformation.Properties.html#2123" class="Bound">α</a>
<a id="2448" class="Keyword">module</a> <a id="2455" href="Categories.NaturalTransformation.Properties.html#2455" class="Module">_</a> <a id="2457" class="Symbol">(</a><a id="2458" href="Categories.NaturalTransformation.Properties.html#2458" class="Bound">F</a> <a id="2460" class="Symbol">:</a> <a id="2462" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="2472" href="Categories.NaturalTransformation.Properties.html#783" class="Generalizable">C</a> <a id="2474" href="Categories.NaturalTransformation.Properties.html#785" class="Generalizable">D</a> <a id="2476" href="Categories.NaturalTransformation.Properties.html#787" class="Generalizable">E</a><a id="2477" class="Symbol">)</a> <a id="2479" class="Keyword">where</a>
<a id="2488" class="Comment">-- there is natural transformation between two partially applied bifunctors.</a>
<a id="2568" href="Categories.NaturalTransformation.Properties.html#2568" class="Function">appˡ-nat</a> <a id="2577" class="Symbol">:</a> <a id="2579" class="Symbol">∀</a> <a id="2581" class="Symbol">{</a><a id="2582" href="Categories.NaturalTransformation.Properties.html#2582" class="Bound">X</a> <a id="2584" href="Categories.NaturalTransformation.Properties.html#2584" class="Bound">Y</a><a id="2585" class="Symbol">}</a> <a id="2587" class="Symbol">→</a> <a id="2589" class="Symbol">(</a><a id="2590" href="Categories.NaturalTransformation.Properties.html#2590" class="Bound">f</a> <a id="2592" class="Symbol">:</a> <a id="2594" href="Categories.Category.Core.html#575" class="Field Operator">Category._⇒_</a> <a id="2607" href="Categories.NaturalTransformation.Properties.html#2472" class="Bound">C</a> <a id="2609" href="Categories.NaturalTransformation.Properties.html#2582" class="Bound">X</a> <a id="2611" href="Categories.NaturalTransformation.Properties.html#2584" class="Bound">Y</a><a id="2612" class="Symbol">)</a> <a id="2614" class="Symbol">→</a> <a id="2616" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="2638" class="Symbol">(</a><a id="2639" href="Categories.Functor.Bifunctor.html#874" class="Function">appˡ</a> <a id="2644" href="Categories.NaturalTransformation.Properties.html#2458" class="Bound">F</a> <a id="2646" href="Categories.NaturalTransformation.Properties.html#2582" class="Bound">X</a><a id="2647" class="Symbol">)</a> <a id="2649" class="Symbol">(</a><a id="2650" href="Categories.Functor.Bifunctor.html#874" class="Function">appˡ</a> <a id="2655" href="Categories.NaturalTransformation.Properties.html#2458" class="Bound">F</a> <a id="2657" href="Categories.NaturalTransformation.Properties.html#2584" class="Bound">Y</a><a id="2658" class="Symbol">)</a>
<a id="2662" href="Categories.NaturalTransformation.Properties.html#2568" class="Function">appˡ-nat</a> <a id="2671" href="Categories.NaturalTransformation.Properties.html#2671" class="Bound">f</a> <a id="2673" class="Symbol">=</a> <a id="2675" href="Categories.NaturalTransformation.Properties.html#2458" class="Bound">F</a> <a id="2677" href="Categories.NaturalTransformation.Core.html#3439" class="Function Operator">∘ˡ</a> <a id="2680" class="Symbol">(</a><a id="2681" href="Categories.Functor.Construction.Constant.html#964" class="Function">constNat</a> <a id="2690" href="Categories.NaturalTransformation.Properties.html#2671" class="Bound">f</a> <a id="2692" href="Categories.Category.Product.html#2966" class="Function Operator">※ⁿ</a> <a id="2695" href="Categories.NaturalTransformation.Properties.html#557" class="Function">idN</a><a id="2698" class="Symbol">)</a>
<a id="2703" href="Categories.NaturalTransformation.Properties.html#2703" class="Function">appʳ-nat</a> <a id="2712" class="Symbol">:</a> <a id="2714" class="Symbol">∀</a> <a id="2716" class="Symbol">{</a><a id="2717" href="Categories.NaturalTransformation.Properties.html#2717" class="Bound">X</a> <a id="2719" href="Categories.NaturalTransformation.Properties.html#2719" class="Bound">Y</a><a id="2720" class="Symbol">}</a> <a id="2722" class="Symbol">→</a> <a id="2724" class="Symbol">(</a><a id="2725" href="Categories.NaturalTransformation.Properties.html#2725" class="Bound">f</a> <a id="2727" class="Symbol">:</a> <a id="2729" href="Categories.Category.Core.html#575" class="Field Operator">Category._⇒_</a> <a id="2742" href="Categories.NaturalTransformation.Properties.html#2474" class="Bound">D</a> <a id="2744" href="Categories.NaturalTransformation.Properties.html#2717" class="Bound">X</a> <a id="2746" href="Categories.NaturalTransformation.Properties.html#2719" class="Bound">Y</a><a id="2747" class="Symbol">)</a> <a id="2749" class="Symbol">→</a> <a id="2751" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="2773" class="Symbol">(</a><a id="2774" href="Categories.Functor.Bifunctor.html#938" class="Function">appʳ</a> <a id="2779" href="Categories.NaturalTransformation.Properties.html#2458" class="Bound">F</a> <a id="2781" href="Categories.NaturalTransformation.Properties.html#2717" class="Bound">X</a><a id="2782" class="Symbol">)</a> <a id="2784" class="Symbol">(</a><a id="2785" href="Categories.Functor.Bifunctor.html#938" class="Function">appʳ</a> <a id="2790" href="Categories.NaturalTransformation.Properties.html#2458" class="Bound">F</a> <a id="2792" href="Categories.NaturalTransformation.Properties.html#2719" class="Bound">Y</a><a id="2793" class="Symbol">)</a>
<a id="2797" href="Categories.NaturalTransformation.Properties.html#2703" class="Function">appʳ-nat</a> <a id="2806" href="Categories.NaturalTransformation.Properties.html#2806" class="Bound">f</a> <a id="2808" class="Symbol">=</a> <a id="2810" href="Categories.NaturalTransformation.Properties.html#2458" class="Bound">F</a> <a id="2812" href="Categories.NaturalTransformation.Core.html#3439" class="Function Operator">∘ˡ</a> <a id="2815" class="Symbol">(</a><a id="2816" href="Categories.NaturalTransformation.Properties.html#557" class="Function">idN</a> <a id="2820" href="Categories.Category.Product.html#2966" class="Function Operator">※ⁿ</a> <a id="2823" href="Categories.Functor.Construction.Constant.html#964" class="Function">constNat</a> <a id="2832" href="Categories.NaturalTransformation.Properties.html#2806" class="Bound">f</a><a id="2833" class="Symbol">)</a>
<a id="2836" class="Keyword">module</a> <a id="2843" href="Categories.NaturalTransformation.Properties.html#2843" class="Module">_</a> <a id="2845" class="Symbol">{</a><a id="2846" href="Categories.NaturalTransformation.Properties.html#2846" class="Bound">F</a> <a id="2848" href="Categories.NaturalTransformation.Properties.html#2848" class="Bound">G</a> <a id="2850" class="Symbol">:</a> <a id="2852" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="2862" href="Categories.NaturalTransformation.Properties.html#783" class="Generalizable">C</a> <a id="2864" href="Categories.NaturalTransformation.Properties.html#785" class="Generalizable">D</a> <a id="2866" href="Categories.NaturalTransformation.Properties.html#787" class="Generalizable">E</a><a id="2867" class="Symbol">}</a> <a id="2869" class="Symbol">(</a><a id="2870" href="Categories.NaturalTransformation.Properties.html#2870" class="Bound">α</a> <a id="2872" class="Symbol">:</a> <a id="2874" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="2896" href="Categories.NaturalTransformation.Properties.html#2846" class="Bound">F</a> <a id="2898" href="Categories.NaturalTransformation.Properties.html#2848" class="Bound">G</a><a id="2899" class="Symbol">)</a> <a id="2901" class="Keyword">where</a>
<a id="2909" class="Keyword">private</a>
<a id="2921" class="Keyword">module</a> <a id="2928" href="Categories.NaturalTransformation.Properties.html#2928" class="Module">C</a> <a id="2930" class="Symbol">=</a> <a id="2932" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="2941" href="Categories.NaturalTransformation.Properties.html#2862" class="Bound">C</a>
<a id="2947" class="Keyword">module</a> <a id="2954" href="Categories.NaturalTransformation.Properties.html#2954" class="Module">D</a> <a id="2956" class="Symbol">=</a> <a id="2958" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="2967" href="Categories.NaturalTransformation.Properties.html#2864" class="Bound">D</a>
<a id="2973" class="Keyword">module</a> <a id="2980" href="Categories.NaturalTransformation.Properties.html#2980" class="Module">E</a> <a id="2982" class="Symbol">=</a> <a id="2984" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="2993" href="Categories.NaturalTransformation.Properties.html#2866" class="Bound">E</a>
<a id="2998" href="Categories.NaturalTransformation.Properties.html#2998" class="Function">appˡ′</a> <a id="3004" class="Symbol">:</a> <a id="3006" class="Symbol">∀</a> <a id="3008" href="Categories.NaturalTransformation.Properties.html#3008" class="Bound">X</a> <a id="3010" class="Symbol">→</a> <a id="3012" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="3034" class="Symbol">(</a><a id="3035" href="Categories.Functor.Bifunctor.html#874" class="Function">appˡ</a> <a id="3040" href="Categories.NaturalTransformation.Properties.html#2846" class="Bound">F</a> <a id="3042" href="Categories.NaturalTransformation.Properties.html#3008" class="Bound">X</a><a id="3043" class="Symbol">)</a> <a id="3045" class="Symbol">(</a><a id="3046" href="Categories.Functor.Bifunctor.html#874" class="Function">appˡ</a> <a id="3051" href="Categories.NaturalTransformation.Properties.html#2848" class="Bound">G</a> <a id="3053" href="Categories.NaturalTransformation.Properties.html#3008" class="Bound">X</a><a id="3054" class="Symbol">)</a>
<a id="3058" href="Categories.NaturalTransformation.Properties.html#2998" class="Function">appˡ′</a> <a id="3064" href="Categories.NaturalTransformation.Properties.html#3064" class="Bound">X</a> <a id="3066" class="Symbol">=</a> <a id="3068" href="Categories.NaturalTransformation.Properties.html#2870" class="Bound">α</a> <a id="3070" href="Categories.NaturalTransformation.Core.html#2919" class="Function Operator">∘ₕ</a> <a id="3073" href="Categories.NaturalTransformation.Properties.html#557" class="Function">idN</a>
<a id="3080" href="Categories.NaturalTransformation.Properties.html#3080" class="Function">appʳ′</a> <a id="3086" class="Symbol">:</a> <a id="3088" class="Symbol">∀</a> <a id="3090" href="Categories.NaturalTransformation.Properties.html#3090" class="Bound">X</a> <a id="3092" class="Symbol">→</a> <a id="3094" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="3116" class="Symbol">(</a><a id="3117" href="Categories.Functor.Bifunctor.html#938" class="Function">appʳ</a> <a id="3122" href="Categories.NaturalTransformation.Properties.html#2846" class="Bound">F</a> <a id="3124" href="Categories.NaturalTransformation.Properties.html#3090" class="Bound">X</a><a id="3125" class="Symbol">)</a> <a id="3127" class="Symbol">(</a><a id="3128" href="Categories.Functor.Bifunctor.html#938" class="Function">appʳ</a> <a id="3133" href="Categories.NaturalTransformation.Properties.html#2848" class="Bound">G</a> <a id="3135" href="Categories.NaturalTransformation.Properties.html#3090" class="Bound">X</a><a id="3136" class="Symbol">)</a>
<a id="3140" href="Categories.NaturalTransformation.Properties.html#3080" class="Function">appʳ′</a> <a id="3146" href="Categories.NaturalTransformation.Properties.html#3146" class="Bound">X</a> <a id="3148" class="Symbol">=</a> <a id="3150" href="Categories.NaturalTransformation.Properties.html#2870" class="Bound">α</a> <a id="3152" href="Categories.NaturalTransformation.Core.html#2919" class="Function Operator">∘ₕ</a> <a id="3155" href="Categories.NaturalTransformation.Properties.html#557" class="Function">idN</a>
<a id="3160" class="Comment">-- unlift universe level</a>
<a id="3185" class="Keyword">module</a> <a id="3192" href="Categories.NaturalTransformation.Properties.html#3192" class="Module">_</a> <a id="3194" class="Symbol">{</a><a id="3195" href="Categories.NaturalTransformation.Properties.html#3195" class="Bound">c</a> <a id="3197" href="Categories.NaturalTransformation.Properties.html#3197" class="Bound">ℓ</a> <a id="3199" href="Categories.NaturalTransformation.Properties.html#3199" class="Bound">ℓ′</a> <a id="3202" href="Categories.NaturalTransformation.Properties.html#3202" class="Bound">e</a><a id="3203" class="Symbol">}</a> <a id="3205" class="Symbol">{</a><a id="3206" href="Categories.NaturalTransformation.Properties.html#3206" class="Bound">F</a> <a id="3208" href="Categories.NaturalTransformation.Properties.html#3208" class="Bound">G</a> <a id="3210" class="Symbol">:</a> <a id="3212" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="3220" href="Categories.NaturalTransformation.Properties.html#783" class="Generalizable">C</a> <a id="3222" class="Symbol">(</a><a id="3223" href="Categories.Category.Instance.Setoids.html#316" class="Function">Setoids</a> <a id="3231" href="Categories.NaturalTransformation.Properties.html#3195" class="Bound">c</a> <a id="3233" href="Categories.NaturalTransformation.Properties.html#3197" class="Bound">ℓ</a><a id="3234" class="Symbol">)}</a> <a id="3237" class="Symbol">(</a><a id="3238" href="Categories.NaturalTransformation.Properties.html#3238" class="Bound">α</a> <a id="3240" class="Symbol">:</a> <a id="3242" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="3264" class="Symbol">(</a><a id="3265" href="Categories.Functor.Construction.LiftSetoids.html#957" class="Function">LiftSetoids</a> <a id="3277" href="Categories.NaturalTransformation.Properties.html#3199" class="Bound">ℓ′</a> <a id="3280" href="Categories.NaturalTransformation.Properties.html#3202" class="Bound">e</a> <a id="3282" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="3285" href="Categories.NaturalTransformation.Properties.html#3206" class="Bound">F</a><a id="3286" class="Symbol">)</a> <a id="3288" class="Symbol">(</a><a id="3289" href="Categories.Functor.Construction.LiftSetoids.html#957" class="Function">LiftSetoids</a> <a id="3301" href="Categories.NaturalTransformation.Properties.html#3199" class="Bound">ℓ′</a> <a id="3304" href="Categories.NaturalTransformation.Properties.html#3202" class="Bound">e</a> <a id="3306" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="3309" href="Categories.NaturalTransformation.Properties.html#3208" class="Bound">G</a><a id="3310" class="Symbol">))</a> <a id="3313" class="Keyword">where</a>
<a id="3321" class="Keyword">open</a> <a id="3326" href="Categories.NaturalTransformation.Core.html#466" class="Module">NaturalTransformation</a> <a id="3348" href="Categories.NaturalTransformation.Properties.html#3238" class="Bound">α</a>
<a id="3352" class="Keyword">open</a> <a id="3357" href="Function.Equality.html#898" class="Module">Π</a>
<a id="3362" href="Categories.NaturalTransformation.Properties.html#3362" class="Function">unlift-nat</a> <a id="3373" class="Symbol">:</a> <a id="3375" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="3397" href="Categories.NaturalTransformation.Properties.html#3206" class="Bound">F</a> <a id="3399" href="Categories.NaturalTransformation.Properties.html#3208" class="Bound">G</a>
<a id="3403" href="Categories.NaturalTransformation.Properties.html#3362" class="Function">unlift-nat</a> <a id="3414" class="Symbol">=</a> <a id="3416" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="3425" class="Keyword">record</a>
<a id="3436" class="Symbol">{</a> <a id="3438" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="3446" class="Symbol">=</a> <a id="3448" class="Symbol">λ</a> <a id="3450" href="Categories.NaturalTransformation.Properties.html#3450" class="Bound">X</a> <a id="3452" class="Symbol">→</a> <a id="3454" class="Keyword">record</a>
<a id="3467" class="Symbol">{</a> <a id="3469" href="Function.Equality.html#1073" class="Field Operator">_⟨$⟩_</a> <a id="3475" class="Symbol">=</a> <a id="3477" class="Symbol">λ</a> <a id="3479" href="Categories.NaturalTransformation.Properties.html#3479" class="Bound">x</a> <a id="3481" class="Symbol">→</a> <a id="3483" href="Level.html#479" class="Field">lower</a> <a id="3489" class="Symbol">(</a><a id="3490" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="3492" href="Categories.NaturalTransformation.Properties.html#3450" class="Bound">X</a> <a id="3494" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="3498" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="3503" href="Categories.NaturalTransformation.Properties.html#3479" class="Bound">x</a><a id="3504" class="Symbol">)</a>
<a id="3512" class="Symbol">;</a> <a id="3514" href="Function.Equality.html#1140" class="Field">cong</a> <a id="3519" class="Symbol">=</a> <a id="3521" class="Symbol">λ</a> <a id="3523" href="Categories.NaturalTransformation.Properties.html#3523" class="Bound">eq</a> <a id="3526" class="Symbol">→</a> <a id="3528" href="Level.html#479" class="Field">lower</a> <a id="3534" class="Symbol">(</a><a id="3535" href="Function.Equality.html#1140" class="Field">cong</a> <a id="3540" class="Symbol">(</a><a id="3541" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="3543" href="Categories.NaturalTransformation.Properties.html#3450" class="Bound">X</a><a id="3544" class="Symbol">)</a> <a id="3546" class="Symbol">(</a><a id="3547" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="3552" href="Categories.NaturalTransformation.Properties.html#3523" class="Bound">eq</a><a id="3554" class="Symbol">))</a>
<a id="3563" class="Symbol">}</a>
<a id="3569" class="Symbol">;</a> <a id="3571" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="3579" class="Symbol">=</a> <a id="3581" class="Symbol">λ</a> <a id="3583" href="Categories.NaturalTransformation.Properties.html#3583" class="Bound">f</a> <a id="3585" href="Categories.NaturalTransformation.Properties.html#3585" class="Bound">eq</a> <a id="3588" class="Symbol">→</a> <a id="3590" href="Level.html#479" class="Field">lower</a> <a id="3596" class="Symbol">(</a><a id="3597" href="Categories.NaturalTransformation.Core.html#827" class="Field">commute</a> <a id="3605" href="Categories.NaturalTransformation.Properties.html#3583" class="Bound">f</a> <a id="3607" class="Symbol">(</a><a id="3608" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="3613" href="Categories.NaturalTransformation.Properties.html#3585" class="Bound">eq</a><a id="3615" class="Symbol">))</a>
<a id="3622" class="Symbol">}</a>
</pre></body></html>
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