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  "body": "\n## Definition\n\nLet $J$ and $\\mathcal{C}$ be [[categories|A2DE66937D]] and let $D \\colon J \\to \\mathcal{C}$ be a [[diagram|HJK75HDWFE]].\nA *limit* of $D$ is a [[cone|7TPBQPPN97]] $(L, \\varphi)$ over $D$, with $\\varphi_j \\colon L \\to D(j)$, that satisfies the following condition.\nFor every cone $(N, \\psi)$ over $D$ there is exactly one morphism $h \\colon N \\to L$ with $\\varphi_j \\circ h = \\psi_j$ for every object $j$ of $J$.\nThe object $L$ is written $\\lim D$ or $\\varprojlim D$, and the $\\varphi_j$ are its *projections*.\n\n## Remarks\n\nEquivalently, a limit of $D$ is a [[terminal object|EK85VSSRMR]] of the category of cones over $D$.\nSo a limit is unique up to a unique isomorphism of cones.\nWhen $J$ is empty, a cone is just an object, and a limit of the empty diagram is a terminal object of $\\mathcal{C}$.\n\nExample: in the category of sets, every diagram $D$ with $J$ small has a limit.\nIt is the subset\n$$L = \\set{(x_j)_j \\in \\textstyle\\prod_j D(j) : D(u)(x_j) = x_k \\text{ for all } u \\colon j \\to k}$$\nof the [[Cartesian product|09CEDJYBKP]] of the sets $D(j)$, with $\\varphi_j$ the coordinate projections.\nA cone $(N, \\psi)$ factors through $L$ by $h(n) = (\\psi_j(n))_j$, which lies in $L$ by the cone condition, and $h$ is forced by $\\varphi_j \\circ h = \\psi_j$.\n\nNon-example: take the diagram of two non-empty sets $X$ and $Y$ with no arrows.\nThe cone $X \\times Y \\times \\{0, 1\\}$ with the projections to $X$ and $Y$ is not a limit.\nThe cone $X \\times Y$ factors through it in two ways, by $(x, y) \\mapsto (x, y, 0)$ and $(x, y) \\mapsto (x, y, 1)$, so uniqueness fails.\n",
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