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  "body": "\n## Definition\n\nLet $F, G \\colon \\mathcal{C} \\to \\mathcal{D}$ be [[functors|28NQ0KRHD3]].\nA *natural transformation* $\\alpha \\colon F \\Rightarrow G$ is a family of [[morphisms|859F0TFVA3]] $\\alpha_A \\colon F(A) \\to G(A)$ of $\\mathcal{D}$, one for each object $A$ of $\\mathcal{C}$.\nThe family must satisfy\n$$G(f) \\circ \\alpha_A = \\alpha_B \\circ F(f)$$\nfor every morphism $f \\colon A \\to B$ of $\\mathcal{C}$.\nThe morphism $\\alpha_A$ is the *component* of $\\alpha$ at $A$.\nIf every component is an isomorphism, $\\alpha$ is a *natural isomorphism*, and $F$ and $G$ are *naturally isomorphic*, written $F \\cong G$.\n\n## Remarks\n\nThe condition says that the naturality square with sides $F(f)$, $\\alpha_B$, $\\alpha_A$ and $G(f)$ is a [[commutative diagram|WV898P4X16]].\nExample: on [[vector spaces|000AYPA9AJ]] over a field $K$, let $\\mathrm{ev}_V \\colon V \\to V^{**}$ send $v$ to the map $\\varphi \\mapsto \\varphi(v)$, where $V^*$ is the [[dual space|07JBSZE0V4]].\nFor a linear map $f \\colon V \\to W$ and $\\psi \\in W^*$, both $f^{**}(\\mathrm{ev}_V(v))$ and $\\mathrm{ev}_W(f(v))$ send $\\psi$ to $\\psi(f(v))$.\nSo $\\mathrm{ev}$ is a natural transformation from the identity functor to the double dual functor.\nOn finite-dimensional spaces it is a natural isomorphism, by the [[canonical isomorphism with the double dual|0ZJM9D7NPS]].\nNon-example: on the category of nonempty sets, choose for each set $X$ a point $x_X \\in X$ and let $\\alpha_X \\colon X \\to X$ be the constant map with value $x_X$.\nThis family is not a natural transformation from the identity functor to itself.\nFor a two-element set $Y$ and the constant map $f \\colon X \\to Y$ whose value is not $x_Y$, the condition would require $f(x_X) = x_Y$.\n",
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