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  "body": "\n## Definition\n\nLet $G \\colon \\mathcal{C} \\to \\mathcal{D}$ be a [[functor|28NQ0KRHD3]], and let $F \\colon J \\to \\mathcal{C}$ be a [[diagram|HJK75HDWFE]].\nFor a [[cone|7TPBQPPN97]] $(L, (\\lambda_j)_{j \\in J})$ over $F$, the image $(G L, (G \\lambda_j)_{j \\in J})$ is a cone over $G \\circ F$.\n$G$ *preserves the limit* of $F$ if for every [[limit|CHS5AFJGWE]] cone $(L, \\lambda)$ of $F$, the cone $(G L, G \\lambda)$ is a limit cone of $G \\circ F$.\n$G$ *preserves limits* if it preserves the limit of every diagram $F \\colon J \\to \\mathcal{C}$ with $J$ a small category.\n\n## Remarks\n\nA functor that preserves limits is also called *continuous*.\nDually, $G$ preserves colimits if it sends colimit cones to colimit cones.\nSince limits are unique up to isomorphism, it suffices to check one limit cone of each diagram.\n\nExample: the forgetful functor $U$ from groups to sets preserves products.\nThe product of groups $H \\times K$ has underlying set $U H \\times U K$, and the projections are the projections of sets.\n\nNon-example: the functor $X \\mapsto X \\sqcup \\set{*}$ from sets to sets does not preserve the terminal object.\nThe terminal object is a one-point set, which is the limit of the empty diagram, and its image has two points.\n\nA related notion goes the other way.\n$G$ *reflects* limits if every cone over $F$ whose image under $G$ is a limit cone is itself a limit cone.\n",
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