Additive functor

Remarks

An additive functor sends zero morphisms to zero morphisms and zero objects to zero objects. If $\mathcal{C}$ and $\mathcal{D}$ are additive categories, a functor is additive if and only if it preserves binary biproducts.

Example: for a fixed abelian group $A$, the functor $\Hom(A, -)$ from abelian groups to abelian groups is additive. For $f, g \colon X \to Y$ and $u \colon A \to X$, one has $(f + g) \circ u = f \circ u + g \circ u$. Likewise the tensor product functor $- \otimes A$ is additive.

Non-example: the functor $X \mapsto X \otimes X$, $f \mapsto f \otimes f$, on abelian groups is not additive. On $X = \Z$, with $\Z \otimes \Z = \Z$, the morphism $\id + \id = 2$ goes to multiplication by $4$, while $F(\id) + F(\id)$ is multiplication by $2$.