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  "body": "\n## Statement\n\nLet $\\mathcal{D}$ be a [[complete category|FWGPKXCPB7]] 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 both of these hold:\n\n1. $G$ [[preserves limits|H4MY8JN8W4]] of small diagrams.\n2. $G$ satisfies the [[solution set condition|CKTDR0VE8D]].\n\n## Remarks\n\nA right adjoint preserves limits, and the unit $\\eta_c$ alone is a solution set, which gives one direction.\nFor the other, fix $c$ and consider the comma category $c \\downarrow G$.\nIt is complete, because $\\mathcal{D}$ is complete and $G$ preserves limits.\nThe product of a solution set is a weakly initial object, and the joint equalizer of all its endomorphisms is an initial object $(F c, \\eta_c)$.\nThese initial objects define the left adjoint $F$.\n\nExample: the categories of groups, rings, modules and other algebraic structures are complete, and their forgetful functors to sets preserve limits.\nA cardinality bound on the subalgebra generated by a set gives the solution set condition.\nSo the theorem gives free algebras without writing them down.\n\nThe solution set condition cannot be dropped.\nThe forgetful functor from complete Boolean algebras to sets preserves limits, but has no left adjoint.\n\n## Explanations\n\n### L1 human\n\nThe theorem gives a test for when a functor has a left adjoint.\nThe necessary condition, that the functor preserve limits, is almost sufficient.\nOne only needs an extra size condition, which says that a set of test objects is enough.\nIt is a general way to show that free objects and similar universal constructions exist.\n\n### L2 human\n\nThe theorem is due to Peter Freyd and is often called the general adjoint functor theorem.\nIn practice, completeness and preservation of limits are easy to check, and the work is in the solution set condition.\nThe special adjoint functor theorem replaces the solution set condition by conditions on $\\mathcal{D}$ alone, such as a cogenerating set.\nIn locally presentable categories the theorem simplifies further: a functor between them is a right adjoint exactly when it preserves limits and is accessible.\n",
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