I am learning Schur's Lemma from page 4 here. It says
Schur's Lemma 1. If $(\rho_1, V_1)$ and $(\rho_2, V_2)$ are irreducible representations of a group $G$, then any nonzero homomorphism $\phi : V_1 \mapsto V_2$ is an isomorphism.
Proof. Assuming $\phi$ is nonzero, we can write $v_2 = \phi(v_1) \in V_2$ for some $v_1 \in V_1$. We can then say that $\rho_2(g)(v_2) = \rho_2(g)(\phi(v_1))$, which by the intertwining property of maps between representations, gives that $\rho_2(g)(v_2) = \phi( \rho_1 (g)(v_1)) \in \phi(V_1)$ ...
Why the $\phi$ can commute with $\rho_1$ and $\rho_2$ in the expression above?