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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] and let $F \\colon \\mathcal{C} \\to \\mathbf{Set}$ be a [[functor|28NQ0KRHD3]] to the category of sets.\nThe *category of elements* $\\int F$, also written $\\mathrm{el}(F)$, is defined as follows:\n\n1. its objects are the pairs $(A, x)$ with $A$ an object of $\\mathcal{C}$ and $x \\in F(A)$;\n2. a morphism $(A, x) \\to (B, y)$ is a morphism $f \\colon A \\to B$ of $\\mathcal{C}$ with $F(f)(x) = y$;\n3. composition and identities are those of $\\mathcal{C}$.\n\n## Remarks\n\nComposition is well defined because $F(g \\circ f)(x) = F(g)(F(f)(x))$.\nIt is the [[comma category|PC6HG1JEGM]] $(\\ast \\downarrow F)$, where $\\ast$ is a one-point set, and the map $(A, x) \\mapsto A$ is a projection functor $\\int F \\to \\mathcal{C}$.\nExample: let $G$ be a [[group|G5S1M9]] seen as a category with one object, and let $F$ be a set $X$ with a [[group action|0WCY42EV9V]] of $G$.\nThen $\\int F$ has the elements of $X$ as objects and a morphism $g \\colon x \\to y$ for each $g \\in G$ with $g \\cdot x = y$; it is the action groupoid.\nNon-example: if $g \\cdot x \\ne y$, then $g$ is not a morphism from $x$ to $y$ in $\\int F$, although it is a morphism of $G$.\nFor a contravariant functor $P \\colon \\mathcal{C}^{\\mathrm{op}} \\to \\mathbf{Set}$, the same construction is usually written with morphisms $f \\colon A \\to B$ such that $P(f)(y) = x$.\n",
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