Adjunction in Abelian Categories

Every adjunction between abelian categories in fact preserves the abelian group structure on the Hom-sets. First, note that $F$ and $G$ will both preserve biproducts (since $F$ preserves colimits and $G$ preserves limits). It follows that $F$ and $G$ preserve the addition operation on Hom-sets, since it can be constructed in terms of biproducts. But now the adjunction bijection $\operatorname{Hom}_D(FX,Y)\to\operatorname{Hom}_C(X,GY)$ is given by taking a morphism $FX\to Y$, applying $G$ to get a morphism $GFX\to GY$, and then composing with the unit of the adjunction $X\to GFX$. Both steps of this preserve addition of morphisms, and hence so does the adjunction bijection.