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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] and let $X$ be an object of $\\mathcal{C}$.\nThe *slice category* $\\mathcal{C}/X$ is defined as follows:\n\n1. its objects are the pairs $(A, h)$ with $h \\colon A \\to X$ a [[morphism|859F0TFVA3]] of $\\mathcal{C}$;\n2. a morphism $(A, h) \\to (A', h')$ is a morphism $f \\colon A \\to A'$ of $\\mathcal{C}$ with $h' \\circ f = h$;\n3. composition and identities are those of $\\mathcal{C}$.\n\n## Remarks\n\nIt is the [[comma category|PC6HG1JEGM]] $(1_{\\mathcal{C}} \\downarrow X)$, where $X$ also denotes the functor from the one-morphism category that picks $X$.\nDually, the *coslice category* $X/\\mathcal{C}$ has as objects the morphisms $X \\to A$, and morphisms $f$ with $f \\circ h = h'$.\nExample: in the category of sets, an object of $\\mathbf{Set}/\\set{0, 1}$ is a set $A$ with a map $h \\colon A \\to \\set{0, 1}$, that is, a set split into two labelled parts.\nWrite $e$ for the identity map of $\\set{0, 1}$; then $e$ is a morphism from $(\\set{0, 1}, e)$ to itself, since $e \\circ e = e$.\nNon-example: let $s \\colon \\set{0, 1} \\to \\set{0, 1}$ swap $0$ and $1$.\nThe identity map $e$ is not a morphism from $(\\set{0, 1}, e)$ to $(\\set{0, 1}, s)$, since $s \\circ e = s \\ne e$.\nThe object $(X, 1_X)$ is a terminal object of $\\mathcal{C}/X$.\n",
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