The following appears in my notes:
Suppose $V$ and $V'$ are vector spaces over a field $F$. Let $G$ be a group, and let $\rho : G \to GL(V)$ and $\rho' : G \to GL(V')$ be representations of $G$. Let $\phi : V \to V'$ be a linear map. We say $\phi$ is a $G$-homomorphism if $ \rho'(g)\circ\phi = \phi\circ\rho(g)$ for all $g$ in $G$. We also say that $\phi$ intertwines $\rho$ and $\rho'$.
$\mathrm{Hom}_G(V,V')$ is the $F$-space of all of these.
It seems strange to me that $\mathrm{Hom}_G(V,V') $ should be independent of $\rho, \rho'$ as the notation suggests. If $\rho$ and $\rho'$ are both the trivial representation, then $\mathrm{Hom}_G(V,V') $ contains all linear maps $V \to V'$. So, does "$\mathrm{Hom}_G(V,V')$" only make sense in the context of specified $\rho, \rho'$, or am I missing something?
Thanks