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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ and $\\mathcal{D}$ be [[preadditive categories|SC6FD5F4J0]].\nA [[functor|28NQ0KRHD3]] $F \\colon \\mathcal{C} \\to \\mathcal{D}$ is *additive* if $F(f + g) = F(f) + F(g)$ for all objects $X, Y$ of $\\mathcal{C}$ and all $f, g \\colon X \\to Y$.\nEquivalently, each map $\\Hom_{\\mathcal{C}}(X, Y) \\to \\Hom_{\\mathcal{D}}(FX, FY)$, $f \\mapsto F(f)$, is a homomorphism of abelian groups.\n\n## Remarks\n\nAn additive functor sends zero morphisms to zero morphisms and zero objects to zero objects.\nIf $\\mathcal{C}$ and $\\mathcal{D}$ are [[additive categories|B2VP96NJSK]], a functor is additive if and only if it preserves binary [[biproducts|BZEJKTQ76T]].\n\nExample: for a fixed [[abelian group|082GHNPJ53]] $A$, the functor $\\Hom(A, -)$ from abelian groups to abelian groups is additive.\nFor $f, g \\colon X \\to Y$ and $u \\colon A \\to X$, one has $(f + g) \\circ u = f \\circ u + g \\circ u$.\nLikewise the [[tensor product|F0HQXQ0HCD]] functor $- \\otimes A$ is additive.\n\nNon-example: the functor $X \\mapsto X \\otimes X$, $f \\mapsto f \\otimes f$, on abelian groups is not additive.\nOn $X = \\Z$, with $\\Z \\otimes \\Z = \\Z$, the morphism $\\id + \\id = 2$ goes to multiplication by $4$, while $F(\\id) + F(\\id)$ is multiplication by $2$.\n",
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