Morphism between representations and equivalent representations

Definition Let $G$ be a finite group. Let $(\phi ,V)$ and $(\theta ,W)$ be two representations. A $G$-map, or morphism of $G$-representations, is a linear map $T:V\to W$ such that for every $g\in G$, $T\circ \phi (g)v=\theta (g)\circ Tv$ That is, the following diagram commutes

V @>{T}>> W \\ @V{\varphi(g)}VV @VV{\theta(g)}V \\ V @>{T}>> W \end{CD} \end{CD}$$ The representations are isomorphic, or equivalent, if $T$ is also a linear isomorphism.