This is my first time dealing with this stuff. $\newcommand{\Hom}{\operatorname{Hom}}$
Assume we have two categories $\mathcal{A}$ and $\mathcal{B}$, and we have a pair $\left (F,G\right)$ of adjoint functors, where $F:\mathcal{A}\to\mathcal{B}$ and $G:\mathcal{B}\to\mathcal{A}$. This means that if $X,X'\in\mathcal{A}$, and $Y,Y'\in\mathcal{B}$, and $f\in\operatorname{Hom}_{\mathcal{A}}\left(X,X'\right)$ and $g\in\operatorname{Hom}_{\mathcal{B}}\left(Y',Y\right)$, then we have a commutative diagram $$ \require{AMScd} \begin{CD} \Hom_\mathcal{A}(X,G(Y)) @>{\beta_{X,Y}}>> \Hom_\mathcal{B}(F(X),Y) \\ @A{G(g) \circ - \circ f}AA @AA{g \circ - \circ F(f)}A \\ \Hom_\mathcal{A}(X',G(Y')) @>>{\beta_{X',Y'}}> \Hom_\mathcal{B}(F(X'),Y') \end{CD} $$ where $\beta_{X,Y}$ and $\beta_{X',Y'}$ are bijection of sets.
My question is the following:
Assume now that, in addition, $\mathcal{A}$ and $\mathcal{B}$ are additive (or better, abelian) categories and that $F$ and $G$ are additive functors. Is it necessarily true that $\beta_{X,Y}$ and $\beta_{X',Y'}$ are group homomorphisms? And if that is not the case, is it true at least that they send $0$ to $0$?