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  "body": "\n## Definition\n\nLet $G \\colon \\mathcal{D} \\to \\mathcal{C}$ be a [[functor|28NQ0KRHD3]] between [[categories|A2DE66937D]].\n$G$ satisfies the *solution set condition* if the following holds for every object $c$ of $\\mathcal{C}$.\nThere are a set $I$, objects $d_i$ of $\\mathcal{D}$ and [[morphisms|859F0TFVA3]] $f_i \\colon c \\to G d_i$ for $i \\in I$ with this property.\nLet $d$ be any object of $\\mathcal{D}$ and $h \\colon c \\to G d$ any morphism.\nThen $h = G(t) \\circ f_i$ for some $i \\in I$ and some morphism $t \\colon d_i \\to d$.\n\n## Remarks\n\nThe family $(f_i)_{i \\in I}$ is called a *solution set* for $c$.\nThe factorization need not be unique, and $I$ must be a set, not a proper class.\n\nExample: let $G$ be the forgetful functor from groups to sets, and let $c = X$ be a set.\nEvery map $h \\colon X \\to G H$ lands in the subgroup $H'$ generated by $h(X)$, which has at most $\\max(\\abs{X}, \\aleph_0)$ elements.\nEach such $H'$ is isomorphic to a group whose underlying set is a subset of $X \\sqcup \\N$.\nAll such groups, with all maps from $X$ into them, form a solution set for $X$.\n\nNon-example: the forgetful functor from complete Boolean algebras to sets fails it, since it preserves limits but has no left adjoint.\nIndeed, there is no free complete Boolean algebra on a countably infinite set.\n",
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