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</pre>-->
<h2 id="summary">Summary</h2>
<p>We work in an ambient category that</p>
<ul>
<li>is extensive (has finite coproducts and pullbacks along
injections)</li>
<li>is cartesian (has finite products, extensive + cartesian also gives
a distributivity isomorphism)</li>
<li>has a parametrized NNO ℕ</li>
<li>has exponentials <code>X^ℕ</code></li>
</ul>
<pre class="Agda"><a id="1642" class="Keyword">module</a> <a id="1649" href="Category.Ambient.html" class="Module">Category.Ambient</a> <a id="1666" class="Keyword">where</a>
<a id="1674" class="Keyword">record</a> <a id="Ambient"></a><a id="1681" href="Category.Ambient.html#1681" class="Record">Ambient</a> <a id="1689" class="Symbol">(</a><a id="1690" href="Category.Ambient.html#1690" class="Bound">o</a> <a id="1692" href="Category.Ambient.html#1692" class="Bound">ℓ</a> <a id="1694" href="Category.Ambient.html#1694" class="Bound">e</a> <a id="1696" class="Symbol">:</a> <a id="1698" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="1703" class="Symbol">)</a> <a id="1705" class="Symbol">:</a> <a id="1707" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1711" class="Symbol">(</a><a id="1712" href="Agda.Primitive.html#931" class="Primitive">suc</a> <a id="1716" href="Category.Ambient.html#1690" class="Bound">o</a> <a id="1718" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1720" href="Agda.Primitive.html#931" class="Primitive">suc</a> <a id="1724" href="Category.Ambient.html#1692" class="Bound">ℓ</a> <a id="1726" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1728" href="Agda.Primitive.html#931" class="Primitive">suc</a> <a id="1732" href="Category.Ambient.html#1694" class="Bound">e</a><a id="1733" class="Symbol">)</a> <a id="1735" class="Keyword">where</a>
<a id="1745" class="Keyword">field</a>
<a id="Ambient.C"></a><a id="1757" href="Category.Ambient.html#1757" class="Field">C</a> <a id="1759" class="Symbol">:</a> <a id="1761" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="1770" href="Category.Ambient.html#1690" class="Bound">o</a> <a id="1772" href="Category.Ambient.html#1692" class="Bound">ℓ</a> <a id="1774" href="Category.Ambient.html#1694" class="Bound">e</a>
<a id="Ambient.extensive"></a><a id="1782" href="Category.Ambient.html#1782" class="Field">extensive</a> <a id="1792" class="Symbol">:</a> <a id="1794" href="Categories.Category.Extensive.html#599" class="Record">Extensive</a> <a id="1804" href="Category.Ambient.html#1757" class="Field">C</a>
<a id="Ambient.cartesian"></a><a id="1812" href="Category.Ambient.html#1812" class="Field">cartesian</a> <a id="1822" class="Symbol">:</a> <a id="1824" href="Categories.Category.Cartesian.html#727" class="Record">Cartesian</a> <a id="1834" href="Category.Ambient.html#1757" class="Field">C</a>
<a id="Ambient.ℕ"></a><a id="1842" href="Category.Ambient.html#1842" class="Field">ℕ</a> <a id="1844" class="Symbol">:</a> <a id="1846" href="Categories.Object.NaturalNumbers.Parametrized.html#2876" class="Record">ParametrizedNNO</a> <a id="1862" class="Symbol">(</a><a id="1863" class="Keyword">record</a> <a id="1870" class="Symbol">{</a> <a id="1872" href="Categories.Category.Cartesian.Bundle.html#488" class="Field">U</a> <a id="1874" class="Symbol">=</a> <a id="1876" href="Category.Ambient.html#1757" class="Field">C</a> <a id="1878" class="Symbol">;</a> <a id="1880" href="Categories.Category.Cartesian.Bundle.html#540" class="Field">cartesian</a> <a id="1890" class="Symbol">=</a> <a id="1892" href="Category.Ambient.html#1812" class="Field">cartesian</a> <a id="1902" class="Symbol">})</a>
<a id="1911" class="Comment">-- _^ℕ : ∀ X → Exponential C (ParametrizedNNO.N ℕ) X</a>
<a id="Ambient.cartesianCategory"></a><a id="1969" href="Category.Ambient.html#1969" class="Function">cartesianCategory</a> <a id="1987" class="Symbol">:</a> <a id="1989" href="Categories.Category.Cartesian.Bundle.html#422" class="Record">CartesianCategory</a> <a id="2007" href="Category.Ambient.html#1690" class="Bound">o</a> <a id="2009" href="Category.Ambient.html#1692" class="Bound">ℓ</a> <a id="2011" href="Category.Ambient.html#1694" class="Bound">e</a>
<a id="2017" href="Category.Ambient.html#1969" class="Function">cartesianCategory</a> <a id="2035" class="Symbol">=</a> <a id="2037" class="Keyword">record</a> <a id="2044" class="Symbol">{</a> <a id="2046" href="Categories.Category.Cartesian.Bundle.html#488" class="Field">U</a> <a id="2048" class="Symbol">=</a> <a id="2050" href="Category.Ambient.html#1757" class="Field">C</a> <a id="2052" class="Symbol">;</a> <a id="2054" href="Categories.Category.Cartesian.Bundle.html#540" class="Field">cartesian</a> <a id="2064" class="Symbol">=</a> <a id="2066" href="Category.Ambient.html#1812" class="Field">cartesian</a> <a id="2076" class="Symbol">}</a>
<a id="2083" class="Keyword">open</a> <a id="2088" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="2097" href="Category.Ambient.html#1757" class="Field">C</a> <a id="2099" class="Keyword">renaming</a> <a id="2108" class="Symbol">(</a><a id="2109" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="2112" class="Symbol">to</a> <a id="2115" class="Field">idC</a><a id="2118" class="Symbol">)</a> <a id="2120" class="Keyword">public</a>
<a id="2131" class="Keyword">open</a> <a id="2136" href="Categories.Category.Extensive.html#599" class="Module">Extensive</a> <a id="2146" href="Category.Ambient.html#1782" class="Field">extensive</a> <a id="2156" class="Keyword">public</a>
<a id="2167" class="Keyword">open</a> <a id="2172" href="Categories.Category.Cocartesian.html#3385" class="Module">Cocartesian</a> <a id="2184" href="Categories.Category.Extensive.html#734" class="Function">cocartesian</a> <a id="2196" class="Keyword">renaming</a> <a id="2205" class="Symbol">(</a><a id="2206" href="Categories.Category.Cocartesian.html#1714" class="Function">+-unique</a> <a id="2215" class="Symbol">to</a> <a id="2218" class="Function">[]-unique</a><a id="2227" class="Symbol">)</a> <a id="2229" class="Keyword">public</a>
<a id="2240" class="Keyword">open</a> <a id="2245" href="Categories.Category.Cartesian.html#727" class="Module">Cartesian</a> <a id="2255" href="Category.Ambient.html#1812" class="Field">cartesian</a> <a id="2265" class="Keyword">public</a>
<a id="2277" class="Comment">-- some helpers</a>
<a id="Ambient.monoidal"></a><a id="2297" href="Category.Ambient.html#2297" class="Function">monoidal</a> <a id="2306" class="Symbol">:</a> <a id="2308" href="Categories.Category.Monoidal.Core.html#1287" class="Record">Monoidal</a> <a id="2317" href="Category.Ambient.html#1757" class="Field">C</a>
<a id="2323" href="Category.Ambient.html#2297" class="Function">monoidal</a> <a id="2332" class="Symbol">=</a> <a id="2334" href="Categories.Category.Cartesian.Monoidal.html#5400" class="Function">CartesianMonoidal.monoidal</a> <a id="2361" href="Category.Ambient.html#1812" class="Field">cartesian</a>
<a id="Ambient.symmetric"></a><a id="2376" href="Category.Ambient.html#2376" class="Function">symmetric</a> <a id="2386" class="Symbol">:</a> <a id="2388" href="Categories.Category.Monoidal.Symmetric.html#997" class="Record">Symmetric</a> <a id="2398" href="Category.Ambient.html#2297" class="Function">monoidal</a>
<a id="2411" href="Category.Ambient.html#2376" class="Function">symmetric</a> <a id="2421" class="Symbol">=</a> <a id="2423" href="Category.Ambient.html#651" class="Function">symm</a> <a id="2428" href="Category.Ambient.html#1757" class="Field">C</a> <a id="2430" href="Category.Ambient.html#1812" class="Field">cartesian</a>
<a id="Ambient.braided"></a><a id="2445" href="Category.Ambient.html#2445" class="Function">braided</a> <a id="2453" class="Symbol">:</a> <a id="2455" href="Categories.Category.Monoidal.Braided.html#643" class="Record">Braided</a> <a id="2463" href="Category.Ambient.html#2297" class="Function">monoidal</a>
<a id="2476" href="Category.Ambient.html#2445" class="Function">braided</a> <a id="2484" class="Symbol">=</a> <a id="2486" href="Categories.Category.Monoidal.Symmetric.html#1047" class="Field">Symmetric.braided</a> <a id="2504" href="Category.Ambient.html#2376" class="Function">symmetric</a>
<a id="2519" class="Keyword">open</a> <a id="2524" href="Categories.Category.BinaryProducts.html#848" class="Module">BinaryProducts</a> <a id="2539" href="Categories.Category.Cartesian.html#801" class="Function">products</a> <a id="2548" class="Keyword">renaming</a> <a id="2557" class="Symbol">(</a><a id="2558" href="Categories.Object.Product.Core.html#797" class="Function">η</a> <a id="2560" class="Symbol">to</a> <a id="2563" class="Function">⁂-η</a><a id="2566" class="Symbol">;</a> <a id="2568" href="Categories.Object.Product.Core.html#739" class="Function">g-η</a> <a id="2572" class="Symbol">to</a> <a id="2575" class="Function">⁂-g-η</a><a id="2580" class="Symbol">;</a> <a id="2582" href="Categories.Object.Product.Core.html#671" class="Function">unique</a> <a id="2589" class="Symbol">to</a> <a id="2592" class="Function">⟨⟩-unique</a><a id="2601" class="Symbol">;</a> <a id="2603" href="Categories.Object.Product.Core.html#1170" class="Function">unique′</a> <a id="2611" class="Symbol">to</a> <a id="2614" class="Function">⟨⟩-unique′</a><a id="2624" class="Symbol">)</a> <a id="2626" class="Keyword">public</a>
<a id="2637" class="Keyword">open</a> <a id="2642" href="Categories.Object.NaturalNumbers.Parametrized.html#2876" class="Module">ParametrizedNNO</a> <a id="2658" href="Category.Ambient.html#1842" class="Field">ℕ</a> <a id="2660" class="Keyword">public</a> <a id="2667" class="Keyword">renaming</a> <a id="2676" class="Symbol">(</a><a id="2677" href="Categories.Object.NaturalNumbers.Parametrized.html#1350" class="Function">η</a> <a id="2679" class="Symbol">to</a> <a id="2682" class="Function">pnno-η</a><a id="2688" class="Symbol">)</a>
<a id="2695" class="Keyword">open</a> <a id="2700" href="Categories.Category.Cartesian.Monoidal.html#1139" class="Module">CartesianMonoidal</a> <a id="2718" href="Category.Ambient.html#1812" class="Field">cartesian</a> <a id="2728" class="Keyword">using</a> <a id="2734" class="Symbol">(</a><a id="2735" href="Categories.Category.Cartesian.Monoidal.html#1572" class="Function">⊤×A≅A</a><a id="2740" class="Symbol">;</a> <a id="2742" href="Categories.Category.Cartesian.Monoidal.html#1852" class="Function">A×⊤≅A</a><a id="2747" class="Symbol">)</a> <a id="2749" class="Keyword">public</a>
<a id="Ambient.distributive"></a><a id="2762" href="Category.Ambient.html#2762" class="Function">distributive</a> <a id="2775" class="Symbol">:</a> <a id="2777" href="Categories.Category.Distributive.html#712" class="Record">Distributive</a> <a id="2790" href="Category.Ambient.html#1757" class="Field">C</a>
<a id="2796" href="Category.Ambient.html#2762" class="Function">distributive</a> <a id="2809" class="Symbol">=</a> <a id="2811" href="Categories.Category.Extensive.Properties.Distributive.html#1506" class="Function">Extensive×Cartesian⇒Distributive</a> <a id="2844" href="Category.Ambient.html#1757" class="Field">C</a> <a id="2846" href="Category.Ambient.html#1782" class="Field">extensive</a> <a id="2856" href="Category.Ambient.html#1812" class="Field">cartesian</a>
<a id="2870" class="Keyword">open</a> <a id="2875" href="Categories.Category.Distributive.html#712" class="Module">Distributive</a> <a id="2888" href="Category.Ambient.html#2762" class="Function">distributive</a> <a id="2901" class="Keyword">hiding</a> <a id="2908" class="Symbol">(</a><a id="2909" href="Categories.Category.Distributive.html#765" class="Field">cartesian</a><a id="2918" class="Symbol">;</a> <a id="2920" href="Categories.Category.Distributive.html#793" class="Field">cocartesian</a><a id="2931" class="Symbol">)</a> <a id="2933" class="Keyword">public</a>
<a id="2944" class="Keyword">open</a> <a id="2949" href="Categories.Morphism.html" class="Module">M&#39;</a> <a id="2952" href="Category.Ambient.html#1757" class="Field">C</a>
<a id="2958" class="Keyword">open</a> <a id="2963" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="2980" class="Keyword">open</a> <a id="2985" href="Categories.Morphism.Reasoning.html" class="Module">MR&#39;</a> <a id="2989" href="Category.Ambient.html#1757" class="Field">C</a>
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<a id="3011" class="Keyword">open</a> <a id="3016" class="Keyword">import</a> <a id="3023" href="Categories.Category.Distributive.Properties.html" class="Module">Categories.Category.Distributive.Properties</a> <a id="3067" href="Category.Ambient.html#2762" class="Function">distributive</a> <a id="3080" class="Keyword">public</a>
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id="7849" class="Symbol">(</a><a id="7850" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="7854" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="7856" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="7858" class="Symbol">)</a> <a id="7860" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7862" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="7864" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="7867" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7869" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="7872" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="7874" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="7877" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="7879" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="7881" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="7929" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="7932" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="7941" class="Symbol">(</a><a id="7942" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="7950" class="Symbol">(</a><a id="7951" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a> <a id="7960" class="Symbol">(</a><a id="7961" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="7965" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a><a id="7974" class="Symbol">)</a> <a id="7976" class="Symbol">(</a><a id="7977" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="7983" href="Categories.Category.Distributive.Properties.html#827" class="Function">distributeˡ⁻¹-i₁</a><a id="7999" class="Symbol">)))</a> <a id="8003" class="Symbol">(</a><a id="8004" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="8012" class="Symbol">(</a><a id="8013" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a> <a id="8022" class="Symbol">(</a><a id="8023" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="8027" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a><a id="8036" class="Symbol">)</a> <a id="8038" class="Symbol">(</a><a id="8039" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="8045" href="Categories.Category.Distributive.Properties.html#977" class="Function">distributeˡ⁻¹-i₂</a><a id="8061" class="Symbol">)))</a> <a id="8065" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="8074" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="8076" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="8090" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8092" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="8094" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="8098" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8100" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="8103" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8105" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="8108" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="8110" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="8113" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8115" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="8117" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="8120" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8122" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="8125" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="8127" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="8130" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="8132" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="8134" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="8136" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="8150" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8152" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="8154" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="8158" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8160" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="8163" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8165" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="8168" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="8170" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="8173" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8175" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="8177" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="8180" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8182" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="8185" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="8187" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="8190" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="8192" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="8194" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="8278" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="8282" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="8291" class="Symbol">(</a><a id="8292" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="8300" href="Categories.Category.BinaryProducts.html#2599" class="Function">⁂∘⟨⟩</a><a id="8304" class="Symbol">)</a> <a id="8306" class="Symbol">(</a><a id="8307" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="8315" href="Categories.Category.BinaryProducts.html#2599" class="Function">⁂∘⟨⟩</a><a id="8319" class="Symbol">)</a> <a id="8321" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="8330" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="8332" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="8346" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8348" class="Symbol">(</a><a id="8349" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="8353" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="8355" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="8357" class="Symbol">)</a> <a id="8359" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8361" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="8363" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="8366" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8368" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="8371" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="8373" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="8375" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="8378" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8380" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="8383" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="8385" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="8388" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="8390" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="8392" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="8394" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="8408" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8410" class="Symbol">(</a><a id="8411" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="8415" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="8417" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="8419" class="Symbol">)</a> <a id="8421" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8423" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="8425" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="8428" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8430" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="8433" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="8435" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="8437" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="8440" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8442" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="8445" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="8447" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="8450" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="8452" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="8454" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="8534" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="8537" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="8546" class="Symbol">(</a><a id="8547" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="8553" href="Categories.Category.Distributive.Properties.html#827" class="Function">distributeˡ⁻¹-i₁</a><a id="8569" class="Symbol">)</a> <a id="8571" class="Symbol">(</a><a id="8572" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="8578" href="Categories.Category.Distributive.Properties.html#977" class="Function">distributeˡ⁻¹-i₂</a><a id="8594" class="Symbol">)</a> <a id="8596" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="8605" class="Symbol">(</a><a id="8606" href="Categories.Morphism.html#2023" class="Field">_≅_.to</a> <a id="8613" href="Categories.Category.BinaryProducts.html#1183" class="Function">×-assoc</a> <a id="8621" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="8624" href="Categories.Morphism.html#2023" class="Field">_≅_.to</a> <a id="8631" href="Categories.Category.BinaryProducts.html#1183" class="Function">×-assoc</a><a id="8638" class="Symbol">)</a> <a id="8809" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="8813" href="Categories.Morphism.Reasoning.Core.html#6665" class="Function">cancelʳ</a> <a id="8821" class="Symbol">(</a><a id="8822" href="Categories.Morphism.html#1586" class="Function">IsIso.isoˡ</a> <a id="8833" href="Categories.Category.Distributive.html#1042" class="Function">isIsoˡ</a><a id="8839" class="Symbol">)</a> <a id="8841" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="8850" class="Symbol">((</a><a id="8852" href="Categories.Morphism.html#2023" class="Field">_≅_.to</a> <a id="8859" href="Categories.Category.BinaryProducts.html#1183" class="Function">×-assoc</a> <a id="8867" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="8870" href="Categories.Morphism.html#2023" class="Field">_≅_.to</a> <a id="8877" href="Categories.Category.BinaryProducts.html#1183" class="Function">×-assoc</a><a id="8884" class="Symbol">)</a> <a id="8886" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8888" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="8901" class="Symbol">)</a> <a id="8903" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8905" href="Categories.Category.Distributive.html#931" class="Function">distributeˡ</a> <a id="9054" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a><a id="9055" class="Symbol">)</a>
<a id="9062" class="Keyword">module</a> <a id="Ambient.M"></a><a id="9069" href="Category.Ambient.html#9069" class="Module">M</a> <a id="9071" class="Symbol">=</a> <a id="9073" href="Categories.Morphism.html" class="Module">M&#39;</a>
<a id="9080" class="Keyword">module</a> <a id="Ambient.MR"></a><a id="9087" href="Category.Ambient.html#9087" class="Module">MR</a> <a id="9090" class="Symbol">=</a> <a id="9092" href="Categories.Morphism.Reasoning.html" class="Module">MR&#39;</a>
<a id="9100" class="Keyword">module</a> <a id="Ambient.MP"></a><a id="9107" href="Category.Ambient.html#9107" class="Module">MP</a> <a id="9110" class="Symbol">=</a> <a id="9112" href="Categories.Morphism.Properties.html" class="Module">MP&#39;</a>
<a id="9121" class="Comment">-- TODO should be in agda-categories</a>
<a id="Ambient.Kleisli⇒Monad⇒Kleisli"></a><a id="9162" href="Category.Ambient.html#9162" class="Function">Kleisli⇒Monad⇒Kleisli</a> <a id="9184" class="Symbol">:</a> <a id="9186" class="Symbol">∀</a> <a id="9188" class="Symbol">(</a><a id="9189" href="Category.Ambient.html#9189" class="Bound">K</a> <a id="9191" class="Symbol">:</a> <a id="9193" href="Categories.Monad.Construction.Kleisli.html#882" class="Function">KleisliTriple</a> <a id="9207" href="Category.Ambient.html#1757" class="Field">C</a><a id="9208" class="Symbol">)</a> <a id="9210" class="Symbol">{</a><a id="9211" href="Category.Ambient.html#9211" class="Bound">X</a> <a id="9213" href="Category.Ambient.html#9213" class="Bound">Y</a><a id="9214" class="Symbol">}</a> <a id="9216" class="Symbol">(</a><a id="9217" href="Category.Ambient.html#9217" class="Bound">f</a> <a id="9219" class="Symbol">:</a> <a id="9221" href="Category.Ambient.html#9211" class="Bound">X</a> <a id="9223" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="9225" href="Categories.Monad.Relative.html#834" class="Field">RMonad.F₀</a> <a id="9235" href="Category.Ambient.html#9189" class="Bound">K</a> <a id="9237" href="Category.Ambient.html#9213" class="Bound">Y</a><a id="9238" class="Symbol">)</a> <a id="9240" class="Symbol">→</a> <a id="9242" href="Categories.Monad.Relative.html#895" class="Field">RMonad.extend</a> <a id="9256" class="Symbol">(</a><a id="9257" href="Categories.Monad.Construction.Kleisli.html#2851" class="Function">Monad⇒Kleisli</a> <a id="9271" href="Category.Ambient.html#1757" class="Field">C</a> <a id="9273" class="Symbol">(</a><a id="9274" href="Categories.Monad.Construction.Kleisli.html#1000" class="Function">Kleisli⇒Monad</a> <a id="9288" href="Category.Ambient.html#1757" class="Field">C</a> <a id="9290" href="Category.Ambient.html#9189" class="Bound">K</a><a id="9291" class="Symbol">))</a> <a id="9294" href="Category.Ambient.html#9217" class="Bound">f</a> <a id="9296" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="9298" href="Categories.Monad.Relative.html#895" class="Field">RMonad.extend</a> <a id="9312" href="Category.Ambient.html#9189" class="Bound">K</a> <a id="9314" href="Category.Ambient.html#9217" class="Bound">f</a>
<a id="9320" href="Category.Ambient.html#9162" class="Function">Kleisli⇒Monad⇒Kleisli</a> <a id="9342" href="Category.Ambient.html#9342" class="Bound">K</a> <a id="9344" href="Category.Ambient.html#9344" class="Bound">f</a> <a id="9346" class="Symbol">=</a> <a id="9348" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="9361" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="9368" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="9372" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9374" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="9381" class="Symbol">(</a><a id="9382" href="Categories.Monad.Relative.html#857" class="Function">unit</a> <a id="9387" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9389" href="Category.Ambient.html#9344" class="Bound">f</a><a id="9390" class="Symbol">)</a> <a id="9392" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="9395" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="9399" href="Categories.Monad.Relative.html#1078" class="Function">kleisli.assoc</a> <a id="9413" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="9422" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="9429" class="Symbol">(</a><a id="9430" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="9437" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="9441" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9443" href="Categories.Monad.Relative.html#857" class="Function">unit</a> <a id="9448" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9450" href="Category.Ambient.html#9344" class="Bound">f</a><a id="9451" class="Symbol">)</a> <a id="9453" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="9456" href="Categories.Monad.Relative.html#1306" class="Function">extend-≈</a> <a id="9465" class="Symbol">(</a><a id="9466" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="9472" href="Categories.Monad.Relative.html#953" class="Function">kleisli.identityʳ</a><a id="9489" class="Symbol">)</a> <a id="9491" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="9499" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="9506" class="Symbol">(</a><a id="9507" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="9511" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9513" href="Category.Ambient.html#9344" class="Bound">f</a><a id="9514" class="Symbol">)</a> <a id="9530" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="9533" href="Categories.Monad.Relative.html#1306" class="Function">extend-≈</a> <a id="9542" class="Symbol">(</a><a id="9543" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a><a id="9552" class="Symbol">)</a> <a id="9554" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="9562" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="9569" href="Category.Ambient.html#9344" class="Bound">f</a> <a id="9593" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="9601" class="Keyword">where</a>
<a id="9616" class="Keyword">module</a> <a id="9623" href="Category.Ambient.html#9623" class="Module">kleisli</a> <a id="9631" class="Symbol">=</a> <a id="9633" href="Category.Ambient.html#934" class="Module">RMonad</a> <a id="9640" href="Category.Ambient.html#9342" class="Bound">K</a>
<a id="9651" class="Keyword">open</a> <a id="9656" href="Category.Ambient.html#9623" class="Module">kleisli</a> <a id="9664" class="Keyword">using</a> <a id="9670" class="Symbol">(</a><a id="9671" href="Categories.Monad.Relative.html#857" class="Function">unit</a><a id="9675" class="Symbol">;</a> <a id="9677" href="Categories.Monad.Relative.html#895" class="Function">extend</a><a id="9683" class="Symbol">;</a> <a id="9685" href="Categories.Monad.Relative.html#1306" class="Function">extend-≈</a><a id="9693" class="Symbol">)</a>
<a id="Ambient.Monad⇒Kleisli⇒Monad"></a><a id="9699" href="Category.Ambient.html#9699" class="Function">Monad⇒Kleisli⇒Monad</a> <a id="9719" class="Symbol">:</a> <a id="9721" class="Symbol">∀</a> <a id="9723" class="Symbol">(</a><a id="9724" href="Category.Ambient.html#9724" class="Bound">M</a> <a id="9726" class="Symbol">:</a> <a id="9728" href="Categories.Monad.html#454" class="Record">Monad</a> <a id="9734" href="Category.Ambient.html#1757" class="Field">C</a><a id="9735" class="Symbol">)</a> <a id="9737" class="Symbol">{</a><a id="9738" href="Category.Ambient.html#9738" class="Bound">X</a> <a id="9740" href="Category.Ambient.html#9740" class="Bound">Y</a><a id="9741" class="Symbol">}</a> <a id="9743" class="Symbol">(</a><a id="9744" href="Category.Ambient.html#9744" class="Bound">f</a> <a id="9746" class="Symbol">:</a> <a id="9748" href="Category.Ambient.html#9738" class="Bound">X</a> <a id="9750" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="9752" href="Categories.Functor.Core.html#797" class="Function">Monad.F.₀</a> <a id="9762" href="Category.Ambient.html#9724" class="Bound">M</a> <a id="9764" href="Category.Ambient.html#9740" class="Bound">Y</a><a id="9765" class="Symbol">)</a> <a id="9767" class="Symbol">→</a> <a id="9769" href="Categories.Functor.Core.html#806" class="Function">Monad.F.₁</a> <a id="9779" class="Symbol">(</a><a id="9780" href="Categories.Monad.Construction.Kleisli.html#1000" class="Function">Kleisli⇒Monad</a> <a id="9794" href="Category.Ambient.html#1757" class="Field">C</a> <a id="9796" class="Symbol">(</a><a id="9797" href="Categories.Monad.Construction.Kleisli.html#2851" class="Function">Monad⇒Kleisli</a> <a id="9811" href="Category.Ambient.html#1757" class="Field">C</a> <a id="9813" href="Category.Ambient.html#9724" class="Bound">M</a><a id="9814" class="Symbol">))</a> <a id="9817" href="Category.Ambient.html#9744" class="Bound">f</a> <a id="9819" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="9821" href="Categories.Functor.Core.html#806" class="Function">Monad.F.₁</a> <a id="9831" href="Category.Ambient.html#9724" class="Bound">M</a> <a id="9833" href="Category.Ambient.html#9744" class="Bound">f</a>
<a id="9839" href="Category.Ambient.html#9699" class="Function">Monad⇒Kleisli⇒Monad</a> <a id="9859" href="Category.Ambient.html#9859" class="Bound">M</a> <a id="9861" href="Category.Ambient.html#9861" class="Bound">f</a> <a id="9863" class="Symbol">=</a> <a id="9865" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="9878" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="9882" class="Symbol">_</a> <a id="9884" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9886" href="Categories.Functor.Core.html#806" class="Function">F.₁</a> <a id="9890" class="Symbol">(</a><a id="9891" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="9895" class="Symbol">_</a> <a id="9897" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9899" href="Category.Ambient.html#9861" class="Bound">f</a><a id="9900" class="Symbol">)</a> <a id="9902" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="9905" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="9913" href="Categories.Functor.Core.html#565" class="Function">F.homomorphism</a> <a id="9928" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="9937" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="9941" class="Symbol">_</a> <a id="9943" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9945" href="Categories.Functor.Core.html#806" class="Function">F.₁</a> <a id="9949" class="Symbol">(</a><a id="9950" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="9954" class="Symbol">_)</a> <a id="9957" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9959" href="Categories.Functor.Core.html#806" class="Function">F.₁</a> <a id="9963" href="Category.Ambient.html#9861" class="Bound">f</a> <a id="9965" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="9968" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="9976" href="Categories.Monad.html#899" class="Function">monad.identityˡ</a> <a id="9992" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10001" href="Categories.Functor.Core.html#806" class="Function">F.₁</a> <a id="10005" href="Category.Ambient.html#9861" class="Bound">f</a> <a id="10007" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="10015" class="Keyword">where</a>
<a id="10029" class="Keyword">module</a> <a id="10036" href="Category.Ambient.html#10036" class="Module">monad</a> <a id="10042" class="Symbol">=</a> <a id="10044" href="Categories.Monad.html#454" class="Module">Monad</a> <a id="10050" href="Category.Ambient.html#9859" class="Bound">M</a>
<a id="10060" class="Keyword">open</a> <a id="10065" href="Category.Ambient.html#10036" class="Module">monad</a> <a id="10071" class="Keyword">using</a> <a id="10077" class="Symbol">(</a><a id="10078" href="Categories.Monad.html#525" class="Function">F</a><a id="10079" class="Symbol">;</a> <a id="10081" href="Categories.Monad.html#547" class="Function">η</a><a id="10082" class="Symbol">;</a> <a id="10084" href="Categories.Monad.html#583" class="Function">μ</a><a id="10085" class="Symbol">)</a>
<a id="Ambient.F₁⇒extend"></a><a id="10091" href="Category.Ambient.html#10091" class="Function">F₁⇒extend</a> <a id="10101" class="Symbol">:</a> <a id="10103" class="Symbol">∀</a> <a id="10105" class="Symbol">(</a><a id="10106" href="Category.Ambient.html#10106" class="Bound">M</a> <a id="10108" class="Symbol">:</a> <a id="10110" href="Categories.Monad.html#454" class="Record">Monad</a> <a id="10116" href="Category.Ambient.html#1757" class="Field">C</a><a id="10117" class="Symbol">)</a> <a id="10119" class="Symbol">{</a><a id="10120" href="Category.Ambient.html#10120" class="Bound">X</a> <a id="10122" href="Category.Ambient.html#10122" class="Bound">Y</a><a id="10123" class="Symbol">}</a> <a id="10125" class="Symbol">(</a><a id="10126" href="Category.Ambient.html#10126" class="Bound">f</a> <a id="10128" class="Symbol">:</a> <a id="10130" href="Category.Ambient.html#10120" class="Bound">X</a> <a id="10132" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="10134" href="Category.Ambient.html#10122" class="Bound">Y</a><a id="10135" class="Symbol">)</a> <a id="10137" class="Symbol">→</a> <a id="10139" href="Categories.Monad.Relative.html#895" class="Field">RMonad.extend</a> <a id="10153" class="Symbol">(</a><a id="10154" href="Categories.Monad.Construction.Kleisli.html#2851" class="Function">Monad⇒Kleisli</a> <a id="10168" href="Category.Ambient.html#1757" class="Field">C</a> <a id="10170" href="Category.Ambient.html#10106" class="Bound">M</a><a id="10171" class="Symbol">)</a> <a id="10173" class="Symbol">(</a><a id="10174" href="Categories.Monad.Relative.html#857" class="Field">RMonad.unit</a> <a id="10186" class="Symbol">(</a><a id="10187" href="Categories.Monad.Construction.Kleisli.html#2851" class="Function">Monad⇒Kleisli</a> <a id="10201" href="Category.Ambient.html#1757" class="Field">C</a> <a id="10203" href="Category.Ambient.html#10106" class="Bound">M</a><a id="10204" class="Symbol">)</a> <a id="10206" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10208" href="Category.Ambient.html#10126" class="Bound">f</a><a id="10209" class="Symbol">)</a> <a id="10211" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="10213" href="Categories.Functor.Core.html#806" class="Function">Monad.F.₁</a> <a id="10223" href="Category.Ambient.html#10106" class="Bound">M</a> <a id="10225" href="Category.Ambient.html#10126" class="Bound">f</a>
<a id="10231" href="Category.Ambient.html#10091" class="Function">F₁⇒extend</a> <a id="10241" href="Category.Ambient.html#10241" class="Bound">M</a> <a id="10243" href="Category.Ambient.html#10243" class="Bound">f</a> <a id="10245" class="Symbol">=</a> <a id="10247" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="10260" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="10264" class="Symbol">_</a> <a id="10266" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10268" href="Categories.Functor.Core.html#806" class="Function">F.₁</a> <a id="10272" class="Symbol">(</a><a id="10273" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="10277" class="Symbol">_</a> <a id="10279" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10281" href="Category.Ambient.html#10243" class="Bound">f</a><a id="10282" class="Symbol">)</a> <a id="10284" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10287" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="10295" href="Categories.Functor.Core.html#565" class="Function">F.homomorphism</a> <a id="10310" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10319" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="10323" class="Symbol">_</a> <a id="10325" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10327" href="Categories.Functor.Core.html#806" class="Function">F.₁</a> <a id="10331" class="Symbol">(</a><a id="10332" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="10336" class="Symbol">_)</a> <a id="10339" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10341" href="Categories.Functor.Core.html#806" class="Function">F.₁</a> <a id="10345" href="Category.Ambient.html#10243" class="Bound">f</a> <a id="10347" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10350" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="10358" href="Category.Ambient.html#10451" class="Function">m-identityˡ</a> <a id="10370" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10379" href="Categories.Functor.Core.html#806" class="Function">F.₁</a> <a id="10383" href="Category.Ambient.html#10243" class="Bound">f</a> <a id="10385" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="10393" class="Keyword">where</a> <a id="10399" class="Keyword">open</a> <a id="10404" href="Categories.Monad.html#454" class="Module">Monad</a> <a id="10410" href="Category.Ambient.html#10241" class="Bound">M</a> <a id="10412" class="Keyword">using</a> <a id="10418" class="Symbol">(</a><a id="10419" href="Categories.Monad.html#525" class="Field">F</a><a id="10420" class="Symbol">;</a> <a id="10422" href="Categories.Monad.html#547" class="Field">η</a><a id="10423" class="Symbol">;</a> <a id="10425" href="Categories.Monad.html#583" class="Field">μ</a><a id="10426" class="Symbol">)</a> <a id="10428" class="Keyword">renaming</a> <a id="10437" class="Symbol">(</a><a id="10438" href="Categories.Monad.html#899" class="Field">identityˡ</a> <a id="10448" class="Symbol">to</a> <a id="10451" class="Field">m-identityˡ</a><a id="10462" class="Symbol">)</a>
<a id="Ambient.extend∘F₁"></a><a id="10469" href="Category.Ambient.html#10469" class="Function">extend∘F₁</a> <a id="10479" class="Symbol">:</a> <a id="10481" class="Symbol">∀</a> <a id="10483" class="Symbol">(</a><a id="10484" href="Category.Ambient.html#10484" class="Bound">M</a> <a id="10486" class="Symbol">:</a> <a id="10488" href="Categories.Monad.html#454" class="Record">Monad</a> <a id="10494" href="Category.Ambient.html#1757" class="Field">C</a><a id="10495" class="Symbol">)</a> <a id="10497" class="Symbol">{</a><a id="10498" href="Category.Ambient.html#10498" class="Bound">X</a> <a id="10500" href="Category.Ambient.html#10500" class="Bound">Y</a> <a id="10502" href="Category.Ambient.html#10502" class="Bound">Z</a><a id="10503" class="Symbol">}</a> <a id="10505" class="Symbol">(</a><a id="10506" href="Category.Ambient.html#10506" class="Bound">f</a> <a id="10508" class="Symbol">:</a> <a id="10510" href="Category.Ambient.html#10500" class="Bound">Y</a> <a id="10512" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="10514" href="Categories.Functor.Core.html#797" class="Function">Monad.F.₀</a> <a id="10524" href="Category.Ambient.html#10484" class="Bound">M</a> <a id="10526" href="Category.Ambient.html#10502" class="Bound">Z</a><a id="10527" class="Symbol">)</a> <a id="10529" class="Symbol">(</a><a id="10530" href="Category.Ambient.html#10530" class="Bound">g</a> <a id="10532" class="Symbol">:</a> <a id="10534" href="Category.Ambient.html#10498" class="Bound">X</a> <a id="10536" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="10538" href="Category.Ambient.html#10500" class="Bound">Y</a><a id="10539" class="Symbol">)</a> <a id="10541" class="Symbol">→</a> <a id="10543" href="Categories.Monad.Relative.html#895" class="Field">RMonad.extend</a> <a id="10557" class="Symbol">(</a><a id="10558" href="Categories.Monad.Construction.Kleisli.html#2851" class="Function">Monad⇒Kleisli</a> <a id="10572" href="Category.Ambient.html#1757" class="Field">C</a> <a id="10574" href="Category.Ambient.html#10484" class="Bound">M</a><a id="10575" class="Symbol">)</a> <a id="10577" href="Category.Ambient.html#10506" class="Bound">f</a> <a id="10579" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10581" href="Categories.Functor.Core.html#806" class="Function">Monad.F.₁</a> <a id="10591" href="Category.Ambient.html#10484" class="Bound">M</a> <a id="10593" href="Category.Ambient.html#10530" class="Bound">g</a> <a id="10595" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="10597" href="Categories.Monad.Relative.html#895" class="Field">RMonad.extend</a> <a id="10611" class="Symbol">(</a><a id="10612" href="Categories.Monad.Construction.Kleisli.html#2851" class="Function">Monad⇒Kleisli</a> <a id="10626" href="Category.Ambient.html#1757" class="Field">C</a> <a id="10628" href="Category.Ambient.html#10484" class="Bound">M</a><a id="10629" class="Symbol">)</a> <a id="10631" class="Symbol">(</a><a id="10632" href="Category.Ambient.html#10506" class="Bound">f</a> <a id="10634" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10636" href="Category.Ambient.html#10530" class="Bound">g</a><a id="10637" class="Symbol">)</a>
<a id="10643" href="Category.Ambient.html#10469" class="Function">extend∘F₁</a> <a id="10653" href="Category.Ambient.html#10653" class="Bound">M</a> <a id="10655" href="Category.Ambient.html#10655" class="Bound">f</a> <a id="10657" href="Category.Ambient.html#10657" class="Bound">g</a> <a id="10659" class="Symbol">=</a> <a id="10661" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="10674" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="10681" href="Category.Ambient.html#10655" class="Bound">f</a> <a id="10683" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10685" href="Categories.Functor.Core.html#806" class="Function">F.₁</a> <a id="10689" href="Category.Ambient.html#10657" class="Bound">g</a> <a id="10691" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10694" class="Symbol">(</a><a id="10695" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="10703" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="10707" class="Symbol">(</a><a id="10708" href="Category.Ambient.html#10091" class="Function">F₁⇒extend</a> <a id="10718" href="Category.Ambient.html#10653" class="Bound">M</a> <a id="10720" href="Category.Ambient.html#10657" class="Bound">g</a><a id="10721" class="Symbol">))</a> <a id="10724" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10733" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="10740" href="Category.Ambient.html#10655" class="Bound">f</a> <a id="10742" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10744" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="10751" class="Symbol">(</a><a id="10752" href="Categories.Monad.Relative.html#857" class="Function">unit</a> <a id="10757" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10759" href="Category.Ambient.html#10657" class="Bound">g</a><a id="10760" class="Symbol">)</a> <a id="10762" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10765" href="Category.Ambient.html#11011" class="Function">k-sym-assoc</a> <a id="10777" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10786" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="10793" class="Symbol">(</a><a id="10794" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="10801" href="Category.Ambient.html#10655" class="Bound">f</a> <a id="10803" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10805" href="Categories.Monad.Relative.html#857" class="Function">unit</a> <a id="10810" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10812" href="Category.Ambient.html#10657" class="Bound">g</a><a id="10813" class="Symbol">)</a> <a id="10815" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10818" href="Categories.Monad.Relative.html#1306" class="Function">extend-≈</a> <a id="10827" class="Symbol">(</a><a id="10828" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="10834" href="Category.Ambient.html#11037" class="Function">k-identityʳ</a><a id="10845" class="Symbol">)</a> <a id="10847" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10856" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="10863" class="Symbol">(</a><a id="10864" href="Category.Ambient.html#10655" class="Bound">f</a> <a id="10866" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10868" href="Category.Ambient.html#10657" class="Bound">g</a><a id="10869" class="Symbol">)</a> <a id="10871" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="10879" class="Keyword">where</a>
<a id="10894" class="Keyword">open</a> <a id="10899" href="Categories.Monad.html#454" class="Module">Monad</a> <a id="10905" href="Category.Ambient.html#10653" class="Bound">M</a> <a id="10907" class="Keyword">using</a> <a id="10913" class="Symbol">(</a><a id="10914" href="Categories.Monad.html#525" class="Field">F</a><a id="10915" class="Symbol">)</a>
<a id="10925" class="Keyword">open</a> <a id="10930" href="Category.Ambient.html#934" class="Module">RMonad</a> <a id="10937" class="Symbol">(</a><a id="10938" href="Categories.Monad.Construction.Kleisli.html#2851" class="Function">Monad⇒Kleisli</a> <a id="10952" href="Category.Ambient.html#1757" class="Field">C</a> <a id="10954" href="Category.Ambient.html#10653" class="Bound">M</a><a id="10955" class="Symbol">)</a> <a id="10957" class="Keyword">using</a> <a id="10963" class="Symbol">(</a><a id="10964" href="Categories.Monad.Relative.html#895" class="Field">extend</a><a id="10970" class="Symbol">;</a> <a id="10972" href="Categories.Monad.Relative.html#857" class="Field">unit</a><a id="10976" class="Symbol">;</a> <a id="10978" href="Categories.Monad.Relative.html#1306" class="Field">extend-≈</a><a id="10986" class="Symbol">)</a> <a id="10988" class="Keyword">renaming</a> <a id="10997" class="Symbol">(</a><a id="10998" href="Categories.Monad.Relative.html#1190" class="Field">sym-assoc</a> <a id="11008" class="Symbol">to</a> <a id="11011" class="Field">k-sym-assoc</a><a id="11022" class="Symbol">;</a> <a id="11024" href="Categories.Monad.Relative.html#953" class="Field">identityʳ</a> <a id="11034" class="Symbol">to</a> <a id="11037" class="Field">k-identityʳ</a><a id="11048" class="Symbol">)</a>
<a id="Ambient.∇"></a><a id="11055" href="Category.Ambient.html#11055" class="Function">∇</a> <a id="11057" class="Symbol">:</a> <a id="11059" class="Symbol">∀</a> <a id="11061" class="Symbol">{</a><a id="11062" href="Category.Ambient.html#11062" class="Bound">X</a><a id="11063" class="Symbol">}</a> <a id="11065" class="Symbol">→</a> <a id="11067" href="Category.Ambient.html#11062" class="Bound">X</a> <a id="11069" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="11071" href="Category.Ambient.html#11062" class="Bound">X</a> <a id="11073" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="11075" href="Category.Ambient.html#11062" class="Bound">X</a>
<a id="11081" href="Category.Ambient.html#11055" class="Function">∇</a> <a id="11083" class="Symbol">=</a> <a id="11085" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="11087" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="11091" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="11093" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="11097" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a>
</pre>
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