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  "body": "\n## Statement\n\nLet $G \\colon \\mathcal{D} \\to \\mathcal{C}$ be a functor that has a [[left adjoint|TZHF6ZM92S]] $F \\colon \\mathcal{C} \\to \\mathcal{D}$.\nThen $G$ [[preserves|H4MY8JN8W4]] every [[limit|CHS5AFJGWE]] that exists in $\\mathcal{D}$.\nDually, $F$ preserves every colimit that exists in $\\mathcal{C}$.\n\n## Remarks\n\nLet $(L, \\lambda)$ be a limit cone of a diagram $K \\colon J \\to \\mathcal{D}$.\nFor an object $c$, cones from $c$ over $G K$ match, by the adjunction, with cones from $F c$ over $K$.\nThese match with morphisms $F c \\to L$, and so with morphisms $c \\to G L$.\nSo $(G L, G \\lambda)$ is a limit cone.\n\nExample: the forgetful functor from groups to sets has a left adjoint, the free group.\nSo the underlying set of a product or an equalizer of groups is the product or the equalizer of the underlying sets.\nThe free group functor preserves coproducts: the free group on $X \\sqcup Y$ is the free product of the free groups on $X$ and on $Y$.\n\nThe theorem gives a test for non-existence.\nThe forgetful functor from groups to sets does not send the initial trivial group to the initial empty set, so it has no right adjoint.\n\n## Explanations\n\n### L1 human\n\nIf a functor has a left adjoint, it behaves well with products, kernels and all other limits.\nFor example, forgetting the group structure of a product of groups leaves the product of the sets.\nDually, free constructions turn sums into sums.\n\n### L2 human\n\nThis is the most used property of adjoint functors.\nIt explains many familiar facts at once, such as that tensor products distribute over direct sums, since $- \\otimes M$ is left adjoint to $\\Hom(M, -)$.\nIt is also the necessary half of the adjoint functor theorems.\nThose theorems show that, under size conditions, preserving limits is enough for a functor to have a left adjoint.\n",
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