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  "body": "\n## Definition\n\nA *sketch* is a triple $\\mathcal{S} = (\\mathcal{A}, L, C)$ with these parts.\n\n1. $\\mathcal{A}$ is a small [[category|A2DE66937D]].\n2. $L$ is a set of [[cones|7TPBQPPN97]] in $\\mathcal{A}$, each over a small diagram in $\\mathcal{A}$.\n3. $C$ is a set of cocones in $\\mathcal{A}$, each under a small diagram in $\\mathcal{A}$.\n\nLet $\\mathcal{K}$ be a category.\nA *model* of $\\mathcal{S}$ in $\\mathcal{K}$ is a [[functor|28NQ0KRHD3]] $M \\colon \\mathcal{A} \\to \\mathcal{K}$ with two properties.\n$M$ sends each cone in $L$ to a [[limit|CHS5AFJGWE]] cone.\n$M$ sends each cocone in $C$ to a [[colimit|BJS1CGTKQ5]] cocone.\nA morphism of models is a natural transformation.\n\n## Remarks\n\nThe sketch is a *limit sketch* when $C$ is empty.\nSome authors use a directed graph with chosen diagrams instead of the category $\\mathcal{A}$; this gives the same models.\n\nExample: let $\\mathcal{A}$ be the free category on the graph with two objects $X, P$ and three arrows $p_1, p_2, m \\colon P \\to X$.\nLet $L$ contain the single cone $(P; p_1, p_2)$ over the discrete diagram $(X, X)$, and let $C$ be empty.\nA model in sets is a functor $M$ with $M(P) \\cong M(X) \\times M(X)$ through $M(p_1), M(p_2)$.\nThen $M(m)$ is a binary operation on $M(X)$, so models are sets with a binary operation.\n\nNon-example: let $M(X) = M(P) = \\set{0, 1}$ with $M(p_1) = M(p_2) = M(m) = \\id$.\nThis functor is not a model, since $\\set{0, 1}$ with two identity maps is not a product of $\\set{0, 1}$ with itself.\nIndeed, the pair of maps $(\\id, \\text{constant } 0)$ does not factor through it.\n",
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