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  "body": "\n## Definition\n\nLet $J$ and $\\mathcal{C}$ be [[categories|A2DE66937D]].\nA *diagram of shape $J$* in $\\mathcal{C}$ is a [[functor|28NQ0KRHD3]] $D \\colon J \\to \\mathcal{C}$.\nThe category $J$ is called the *index category* or *shape* of the diagram.\nThe diagram is *small* if $J$ is small, and *finite* if $J$ has finitely many objects and morphisms.\n\n## Remarks\n\nThe objects $D(j)$ and morphisms $D(u)$ for $u$ in $J$ form the picture that the diagram draws in $\\mathcal{C}$.\nBecause $D$ is a functor, $D(v \\circ u) = D(v) \\circ D(u)$, so every composite in $J$ is sent to the corresponding composite in $\\mathcal{C}$.\n\nExample: let $J$ have three objects $1, 2, 3$ and, besides identities, morphisms $u \\colon 1 \\to 3$ and $v \\colon 2 \\to 3$.\nA diagram of shape $J$ in the category of sets is a pair of functions $f = D(u) \\colon X \\to Z$ and $g = D(v) \\colon Y \\to Z$; its limit is the pullback of $f$ and $g$.\n\nNon-example: let $J$ have one object $\\ast$ and one non-identity morphism $e$ with $e \\circ e = e$.\nSending $\\ast$ to $\\Z$ and $e$ to the map $n \\mapsto -n$ is not a diagram of shape $J$ in the category of sets.\nThe image of $e \\circ e$ would have to be $n \\mapsto -n$, but the composite of $n \\mapsto -n$ with itself is the identity.\n",
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