Let $U,V$ and $W$ be $kG$-modules. Then, as $kG$-modules, show the following are isomorphic.
$$\text{Hom}_{k}(U \oplus V,W) \cong \text{Hom}_{k}(U,W) \oplus \text{Hom}_{k}(U,W),$$
$$\text{Hom}_{k}(U,V \oplus W) \cong \text{Hom}_{k}(U,V) \oplus \text{Hom}_{k}(U,W).$$
I'm really struggling with this proof. I've been able to verify that the dimensions are the same, but I'm not sure that's enough as these aren't vector spaces. Any hints?
Hint: If $\eta\in \mathrm{Hom}(U\oplus V, W)$, then $\eta(u,v) = \eta(u,0) + \eta(0,v)$.
What can we now define using $\eta(u,0)$ and $\eta(0,v)$?