Definition 12.3.1. A category $\mathcal{A}$ is called *preadditive* if each morphism set $\mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(x, y)$ is endowed with the structure of an abelian group such that the compositions

are bilinear. A functor $F : \mathcal{A} \to \mathcal{B}$ of preadditive categories is called *additive* if and only if $F : \mathop{\mathrm{Mor}}\nolimits (x, y) \to \mathop{\mathrm{Mor}}\nolimits (F(x), F(y))$ is a homomorphism of abelian groups for all $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$.

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