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  "body": "\n## Statement\n\nLet $\\mathcal{A}$ be a small [[abelian category|Y9CZ3MTNF1]], that is, an abelian category whose objects form a set.\nThen there exist a [[ring|0B1261B5ED]] $R$ with identity and a [[functor|28NQ0KRHD3]] $F \\colon \\mathcal{A} \\to R\\text{-}\\mathrm{Mod}$ into the [[category of left $R$-modules|JBSSXP4FFY]] such that:\n\n1. $F$ is [[full and faithful|GRG8WEBSH5]];\n2. $F$ is an [[exact functor|S1G61TDW0K]].\n\n## Remarks\n\nThe idea of the proof is to embed $\\mathcal{A}$, by a Yoneda-type construction, into a category of left exact functors from $\\mathcal{A}$ to abelian groups.\nThat category is abelian and has enough injectives and a generator, which leads to an abelian category with a projective generator $P$ containing a copy of $\\mathcal{A}$.\nOne then takes $R$ to be the endomorphism ring of a suitable such generator and uses the functor $\\Hom(P, -)$.\nSince $F$ is full, faithful and exact, a diagram in $\\mathcal{A}$ commutes, or a sequence in $\\mathcal{A}$ is exact, exactly when its image under $F$ does.\nSo a diagram lemma such as the five lemma or the snake lemma can be proved in an arbitrary abelian category by chasing elements of modules.\nFor a large abelian category, one applies the theorem to a small full abelian subcategory that contains the given diagram and is closed under kernels and cokernels.\n\n## Explanations\n\n### L1 human\n\nAbelian categories are an abstract setting for kernels, cokernels and exact sequences.\nThe theorem says that a small abelian category is, up to a faithful copy, just a part of a category of modules over some ring.\nSo nothing new can happen in an abstract abelian category that does not already happen for modules.\n\n### L2 human\n\nThe main use of the theorem is to justify element chasing.\nA statement about a finite diagram of objects and morphisms in an abelian category can be transferred to modules, proved there with elements, and transferred back.\nThis is why many texts prove diagram lemmas only for modules.\nAn alternative that stays inside the category is the method of [[pseudo-elements|1WV2THA6NY]].\nThe [[Gabriel–Popescu theorem|3MP41FQHZ5]] is a related embedding result for Grothendieck categories, where the functor is not exact but has an exact left adjoint.\n",
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