Let $C,D$ be arbitrary categories and $F:C\to D, G:D\to C$ functors. We say $F,G$ are adjoint if for every $X\in C, Y \in D$ there is an isomorphism of Hom-Sets between $Hom_D(FX,Y)\cong Hom_C(X,GY)$, which is natural in $X$ and $Y$.
If for example $C,D$ are abelian, we have an abelian group structure on the Hom-Sets. Does anything interesting happen if we add the condition that the natural isomorphism must be an isomorphism of abelian groups and not just sets?
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.