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  "body": "\n## Summary\n\nLet $S \\colon \\mathcal{A} \\to \\mathcal{C}$ and $T \\colon \\mathcal{B} \\to \\mathcal{C}$ be [[functors|28NQ0KRHD3]].\nThe comma category $(S \\downarrow T)$ has as objects the morphisms $S(A) \\to T(B)$ of $\\mathcal{C}$.\nIts morphisms are the pairs of morphisms that make the evident square commute.\nIt turns morphisms of one category into objects of a new category.\n[[Slice categories|1TC7W7BQT3]], arrow categories and [[categories of elements|S9EJKKRHW1]] are special comma categories.\n[[Universal properties|EN4DD2FAJH]] and [[adjoint functors|TZHF6ZM92S]] can be expressed as initial or terminal objects of comma categories.\n\n## Explanations\n\n### L1 human\n\nA comma category collects arrows of a category $\\mathcal{C}$ that go from one kind of object to another, and treats each arrow as a new object.\nAn arrow between two such objects is a pair of maps at the two ends that fits with them in a commuting square.\nFor example, taking all arrows of $\\mathcal{C}$ at both ends gives the arrow category, whose objects are morphisms $h \\colon X \\to Y$ of $\\mathcal{C}$.\nIn the arrow category of sets, the map $\\R \\to \\R$, $x \\mapsto x^2$, is one object, and the inclusion $\\Z \\to \\R$ is another.\n",
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