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  "body": "\n## Definition\n\nLet $F \\colon \\mathcal{C} \\to \\mathcal{D}$ be a [[functor|28NQ0KRHD3]].\nFor an object $d$ of $\\mathcal{D}$, let $d \\downarrow F$ be the [[comma category|PC6HG1JEGM]] whose objects are pairs $(c, f)$ with $c$ an object of $\\mathcal{C}$ and $f \\colon d \\to F c$.\nA morphism $(c, f) \\to (c', f')$ in it is a morphism $h \\colon c \\to c'$ with $F(h) \\circ f = f'$.\n$F$ is *final* if for every object $d$ of $\\mathcal{D}$ the category $d \\downarrow F$ is nonempty and connected.\n\n## Remarks\n\nHere a category is connected when any two objects are joined by a finite zigzag of morphisms.\nIf $F$ is final, then for every functor $G \\colon \\mathcal{D} \\to \\mathcal{E}$ the [[colimit|BJS1CGTKQ5]] of $G \\circ F$ exists if and only if the colimit of $G$ exists.\nThe canonical morphism $\\operatorname{colim} (G \\circ F) \\to \\operatorname{colim} G$ is then an isomorphism.\nFor small $\\mathcal{C}$ and $\\mathcal{D}$ this property for all $G$ is equivalent to finality.\n\nExample: let $\\mathcal{D}$ have a terminal object $t$, and let $F$ be the functor from the one-object category picking out $t$.\nFor each $d$ the category $d \\downarrow F$ has one object, the unique arrow $d \\to t$, and only its identity, so it is nonempty and connected.\nHence $\\operatorname{colim} G = G(t)$ for every $G$.\n\nNon-example: let $\\mathcal{D}$ be the poset $\\set{0 \u003c 1}$ and let $F$ pick out $0$.\nThere is no arrow $1 \\to 0$, so $1 \\downarrow F$ is empty and $F$ is not final.\n\nThe inclusion of a cofinal subset of a directed set is a final functor.\n",
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