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  "body": "\n## Statement\n\nLet $\\mathcal{C}$ be a [[Grothendieck category|A7W468BQSM]] and let $U$ be a [[generator|CK08JW32WB]] of $\\mathcal{C}$.\nLet $R = \\End(U)$ be the endomorphism [[ring|0B1261B5ED]] of $U$, so that $\\Hom(U, X)$ is a right $R$-module for each object $X$, with $R$ acting by composition.\nLet $G = \\Hom(U, -) \\colon \\mathcal{C} \\to \\mathrm{Mod}\\text{-}R$ be the [[hom functor|VJXS8AK67E]] into the [[category of right $R$-modules|JBSSXP4FFY]].\nThen:\n\n1. $G$ is [[full and faithful|GRG8WEBSH5]];\n2. $G$ has a [[left adjoint|TZHF6ZM92S]] $T \\colon \\mathrm{Mod}\\text{-}R \\to \\mathcal{C}$;\n3. $T$ is an [[exact functor|S1G61TDW0K]].\n\n## Remarks\n\nThe functor $T$ is the colimit-preserving functor with $T(R) = U$, often written $T(M) = M \\otimes_R U$.\nThe hard part of the proof is the exactness of $T$; it uses that filtered colimits are exact in a Grothendieck category and that $U$ generates.\nAs a consequence, $\\mathcal{C}$ is equivalent to the quotient of $\\mathrm{Mod}\\text{-}R$ by the [[Serre subcategory|CSD6W0JV0P]] of modules $M$ with $T(M) = 0$.\nThat subcategory is closed under arbitrary direct sums, so it is a localizing subcategory.\nExample: for $\\mathcal{C} = \\mathrm{Mod}\\text{-}S$ and $U = S$, one gets $R \\cong S$, the functor $G$ is an equivalence, and the localizing subcategory is $0$.\nExample: the category of abelian sheaves on a topological space is a Grothendieck category, so the theorem presents it as a quotient of a module category.\n\n## Explanations\n\n### L1 human\n\nA Grothendieck category is an abelian category with good infinite colimits and a generating object, like a category of modules or of sheaves.\nThe theorem says that every such category sits inside a category of modules in a very controlled way.\nIt is obtained from the modules by declaring a certain class of modules to be zero.\n\n### L2 human\n\nThe theorem is an analogue of the [[Freyd–Mitchell embedding theorem|8V296XR2M3]] for large categories with a generator.\nThe embedding $G$ is left exact but in general not exact; in exchange, it has an exact left adjoint, so $\\mathcal{C}$ is a reflective subcategory of $\\mathrm{Mod}\\text{-}R$ with exact reflector.\nThis makes it a basic tool for transferring module-theoretic results to Grothendieck categories.\nIt is also the starting point of the description of Grothendieck categories as localizations of module categories.\n",
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