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