Suppose $\mathcal{C},\mathcal{D}$ are additive categories and $\Phi \colon \mathcal{C} \to \mathcal{D}$ is a functor with a right adjoint $\Psi$. Show that $\Phi$ commutes with direct sums and that $\Phi_{x,y} \colon \textrm{Hom}_{\mathcal{C}}(x,y) \to \textrm{Hom}_{\mathcal{D}}(\Phi(x),\Phi(y))$ is a group homomorphism for any objects $x,y$ in $\mathcal{C}$.
I think I've got the part about commuting with direct sums, but I'm not sure about the homomorphism part. I believe that the fact that $\Psi$ is a right adjoint should imply that $\textrm{Hom}_{\mathcal{D}}(\Phi(x),y)$ and $\textrm{Hom}_{\mathcal{C}}(x,\Psi(y))$ are isomorphic as abelian groups. I could thus say that $\textrm{Hom}_{\mathcal{D}}(\Phi(x),\Phi(y))$ and $\textrm{Hom}_{\mathcal{C}}(x,\Psi(\Phi(y)))$ are isomorphic, but this isn't quite what I want as I don't know that $\Psi(\Phi(y)) = y$.