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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nA *diagram* in $\\mathcal{C}$ is a directed graph with each vertex labelled by an object and each edge labelled by a [[morphism|859F0TFVA3]] of $\\mathcal{C}$.\nAn edge from a vertex labelled $A$ to a vertex labelled $B$ must be labelled by a morphism $A \\to B$.\nThe composite of a directed path of edges $e_1, \\dots, e_n$ with $n \\ge 1$ and labels $f_1, \\dots, f_n$ is $f_n \\circ \\dots \\circ f_1$.\nThe diagram *commutes* if any two directed paths of length at least $1$ with the same start and the same end have the same composite.\n\n## Remarks\n\nExample: the square with vertices $A, B, C, D$ and edges $f \\colon A \\to B$, $g \\colon B \\to D$, $h \\colon A \\to C$, $k \\colon C \\to D$ commutes exactly when $g \\circ f = k \\circ h$.\nFor instance, in the category of sets with $A = B = C = D = \\R$, the maps $f = k = (x \\mapsto 2x)$ and $g = h = (x \\mapsto 3x)$ give a commuting square, since both composites are $x \\mapsto 6x$.\nNon-example: with $f = k = (x \\mapsto x + 1)$ and $g = h = (x \\mapsto 2x)$, the square does not commute.\nIndeed $g(f(x)) = 2x + 2$ while $k(h(x)) = 2x + 1$.\nIn practice one only asks for commutativity of the paths that the text names.\nFor a diagram with a loop $f$, the convention above would force $f = f \\circ f$.\nMore formally, a diagram of shape $J$ is a functor from a category $J$ to $\\mathcal{C}$.\n",
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