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  "body": "\n## Definition\n\nLet $S \\colon \\mathcal{A} \\to \\mathcal{C}$ and $T \\colon \\mathcal{B} \\to \\mathcal{C}$ be [[functors|28NQ0KRHD3]].\nThe *comma category* $(S \\downarrow T)$ is defined as follows:\n\n1. its objects are the triples $(A, B, h)$ with $A$ an object of $\\mathcal{A}$, $B$ an object of $\\mathcal{B}$ and $h \\colon S(A) \\to T(B)$ a [[morphism|859F0TFVA3]] of $\\mathcal{C}$;\n2. a morphism $(A, B, h) \\to (A', B', h')$ is a pair $(f, g)$ with $f \\colon A \\to A'$ in $\\mathcal{A}$ and $g \\colon B \\to B'$ in $\\mathcal{B}$ such that $T(g) \\circ h = h' \\circ S(f)$;\n3. composition is $(f', g') \\circ (f, g) = (f' \\circ f, g' \\circ g)$, and the identity of $(A, B, h)$ is $(1_A, 1_B)$.\n\n## Remarks\n\nThe condition in clause 2 says that the square with sides $S(f)$, $h'$, $h$ and $T(g)$ is a [[commutative diagram|WV898P4X16]]; pasting two such squares shows that composites are again morphisms.\nExample: with $S = T = 1_{\\mathcal{C}}$, the comma category is the arrow category of $\\mathcal{C}$, whose objects are the morphisms $h \\colon X \\to Y$.\nIn the arrow category of sets, the pair $(f, g)$ with $f = g = (x \\mapsto -x)$ is a morphism from $h = (x \\mapsto x^3)$ on $\\R$ to itself, since $(-x)^3 = -x^3$.\nNon-example: for $h = (x \\mapsto x^2)$ on $\\R$, the same pair $(f, g)$ is not a morphism from $h$ to $h$, since $-x^2 \\ne (-x)^2$ when $x \\ne 0$.\nWhen $\\mathcal{A}$ is the category with one object and one morphism, $S$ picks an object $X$ of $\\mathcal{C}$ and one writes $(X \\downarrow T)$.\nThere are projection functors from $(S \\downarrow T)$ to $\\mathcal{A}$ and to $\\mathcal{B}$.\n",
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