\newcommand{\sqdiagram}[9]{$\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}$ }
 A functor between two preadditive categories $\displaystyle \mathcal{\A_P}$
and $\displaystyle \mathcal{\A'_P}$

if for any pair of morphisms $\displaystyle f, g \in Hom_{\mathcal{\A_P}}(X,Y)$ one has that ${\displaystyle F(f+g)=F(f)+F(g)}$.