I am trying to prove the following:
Let $\left(p_1, V_1\right),\left(\rho_2, V_2\right)$, and $(\sigma, W)$ be representations of a finite group G. Prove that $\operatorname{Hom}\left(\rho_1 \oplus \rho_2, \sigma\right)=\operatorname{Hom}\left(\rho_1, \sigma\right) \oplus \operatorname{Hom}\left(\rho_2, \sigma\right)$
The direction $\subset$ was easy, but when taking $A\in \operatorname{Hom}\left(\rho_1, \sigma\right) \cap \operatorname{Hom}\left(\rho_2, \sigma\right)$ to show that $A=0$, I have difficulty even with the definition. I know $A\in\operatorname{Hom}(V,W)$ and $A\rho_1(g)=\sigma(g)A$ but can I even write $A\rho_1(g)(v_2)=\sigma(g)A(v_2)$? It seems undefined since $\rho_1(g)\in GL(V_1)$.
Note - we usually only deal with finite dimensional vector spaces, if that matters.