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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a small [[category|A2DE66937D]] and let $\\mathcal{D}$ be a category.\nThe *functor category* $[\\mathcal{C}, \\mathcal{D}]$ is defined as follows:\n\n1. its objects are the [[functors|28NQ0KRHD3]] $\\mathcal{C} \\to \\mathcal{D}$;\n2. its morphisms $F \\to G$ are the [[natural transformations|KYCPCC57FT]] $\\alpha \\colon F \\Rightarrow G$;\n3. the composite of $\\alpha \\colon F \\Rightarrow G$ and $\\beta \\colon G \\Rightarrow H$ is $\\beta \\circ \\alpha$ with components $(\\beta \\circ \\alpha)_A = \\beta_A \\circ \\alpha_A$, and the identity of $F$ has components $1_{F(A)}$.\n\n## Remarks\n\nThe composite $\\beta \\circ \\alpha$ is natural, because both of its naturality squares can be pasted together.\nIt is also written $\\mathcal{D}^{\\mathcal{C}}$ or $\\mathrm{Fun}(\\mathcal{C}, \\mathcal{D})$.\nSmallness of $\\mathcal{C}$ ensures that, when $\\mathcal{D}$ is locally small, the natural transformations between two functors form a set.\nExample: let $G$ be a [[group|G5S1M9]], seen as a category with one object $\\ast$ and with the elements of $G$ as morphisms.\nA functor $G \\to \\mathbf{Set}$ is a set $X = F(\\ast)$ with a [[group action|0WCY42EV9V]] $g \\cdot x = F(g)(x)$, and a natural transformation is a map $h$ with $h(g \\cdot x) = g \\cdot h(x)$.\nSo $[G, \\mathbf{Set}]$ is the category of $G$-sets.\nNon-example: a family of morphisms $\\alpha_A \\colon F(A) \\to G(A)$ that fails one naturality square is not a morphism of $[\\mathcal{C}, \\mathcal{D}]$.\nA morphism of $[\\mathcal{C}, \\mathcal{D}]$ is an isomorphism exactly when all its components are isomorphisms.\n",
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