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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ and $\\mathcal{B}$ be [[abelian categories|Y9CZ3MTNF1]] and let $F \\colon \\mathcal{A} \\to \\mathcal{B}$ be an [[additive functor|KAN07YC9T2]].\nLet $0 \\to A' \\to A \\to A'' \\to 0$ denote an arbitrary [[short exact sequence|KJKNJMGKZZ]] in $\\mathcal{A}$.\n\n1. $F$ is *left exact* if $0 \\to F(A') \\to F(A) \\to F(A'')$ is exact for every such sequence.\n2. $F$ is *right exact* if $F(A') \\to F(A) \\to F(A'') \\to 0$ is exact for every such sequence.\n3. $F$ is *exact* if $0 \\to F(A') \\to F(A) \\to F(A'') \\to 0$ is exact for every such sequence, that is, if $F$ is both left exact and right exact.\n\n## Remarks\n\nAn exact functor preserves all exact sequences, not only the short ones.\nA functor is left exact if and only if it preserves kernels, and right exact if and only if it preserves cokernels.\nFor a contravariant functor the same words are used after replacing $\\mathcal{A}$ by its opposite category.\n\nExample: for a ring $R$ and an $R$-module $M$, the functor $\\Hom_R(M, -)$ is left exact, and $M \\otimes_R -$ is right exact.\nThe functor $\\Q \\otimes_{\\Z} -$ on abelian groups is exact, because $\\Q$ is a flat $\\Z$-module.\n\nNon-example: $F = \\Z/2\\Z \\otimes_{\\Z} -$ is right exact but not exact.\nApply it to $0 \\to \\Z \\xrightarrow{2} \\Z \\to \\Z/2\\Z \\to 0$.\nThe first map becomes multiplication by $2$ on $\\Z/2\\Z$, which is zero and so not injective.\n",
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