Let $\mathcal{A}$ and $\mathcal{B}$ be ab-categories. A functor $F:\mathcal{A}\to \mathcal{B}$ is called an additive functor if, for any objects $A,B$ in $\mathcal{A}$ the function $$F_{(A,B)}: \hom(A,B)\to \hom(F(A),F(B))$$ given by $F_{(A,B)}(f)=F(f)$ is a group homomorphism. In other words, if $f,g: A\to B$ are two morphisms with common domain $A$ and codomain $B$ then $$F(f+g)=F(f)+F(g).$$

For example, the hom functor $\hom(A,-)$ where $A$ is an object in an abelian category, is additive.

Remark. It can be shown that any exact functor between abelian categories is additive.

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