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  "body": "\n## Statement\n\nLet $\\mathcal{D}$ be a [[complete category|FWGPKXCPB7]] whose hom-sets are sets.\nAssume that $\\mathcal{D}$ is [[well-powered|XVG7FPG227]] and has a cogenerating set.\nLet $\\mathcal{C}$ be a category whose hom-sets are sets, and let $G \\colon \\mathcal{D} \\to \\mathcal{C}$ be a functor.\nThen $G$ has a [[left adjoint|TZHF6ZM92S]] if and only if $G$ [[preserves limits|H4MY8JN8W4]] of small diagrams.\n\n## Remarks\n\nHere a set $Q$ of objects is cogenerating when, for any two different morphisms $f, g \\colon a \\to b$, some $q \\colon b \\to x$ with $x \\in Q$ has $q f \\neq q g$.\nThis is the dual of a [[generator|CK08JW32WB]].\nThe proof constructs an initial object of each comma category $c \\downarrow G$.\nLet $P$ be the product of copies of $x$, one for each $x \\in Q$ and each morphism $c \\to G x$.\nThen one intersects all subobjects $S$ of $P$ such that the induced map $c \\to G P$ factors through $G S$.\nWell-poweredness makes this intersection a small limit.\n\nExample: the category of compact Hausdorff spaces is complete and well-powered, and the unit interval $[0, 1]$ is a cogenerator.\nThe inclusion into topological spaces preserves limits.\nSo it has a left adjoint, the [[Stone–Čech compactification|D9MWP5C85W]].\n\n## Explanations\n\n### L1 human\n\nThis theorem is another test for a left adjoint.\nInstead of a size condition on the functor, it asks for size conditions on the category where the functor starts.\nThen preserving limits is all that is needed.\nIts most famous use is to show that every space has a Stone–Čech compactification.\n\n### L2 human\n\nCompared with Freyd's general theorem, the solution set condition is replaced by well-poweredness and a cogenerating set in $\\mathcal{D}$.\nThese conditions concern only $\\mathcal{D}$, so they are checked once for all functors out of $\\mathcal{D}$.\nThe dual theorem gives right adjoints of colimit-preserving functors out of a cocomplete, co-well-powered category with a generating set.\nFor instance, every colimit-preserving functor from a category of presheaves on a small category has a right adjoint.\n",
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