Some properties for morphism of representations

Let $psi : Gto GL(V)$ and $varphi :G to GL (W)$ be representations of finite group $G$. A linear map $T: Vto W$ is called a morphism from $psi$ to $varphi$ if it satisfys $Tpsi_g= varphi_g T$ for each $gin G$ and let $mathrm{Hom}_G (psi ,varphi)$ denote the set of all morphisms. In this paper, we make full stufy of the subspace $mathrm{Hom}_G(psi, varphi)$. As byproducts, we include the proof of the first orthogonality relation and Schurs orthogonality relation.