I am currently self-studying the representation theory of the symmetric group by following the book The Symmetric Group by Bruce E. Sagan. All groups here are finite and the base field is $\mathbb{C}$. I'm having trouble seeing how he arrives at the following proposition:
Proposition 1.7.10. Let $V$ and $W$ be $G$-modules with $V$ irreducible. Then dim Hom$(V,W)$ is the multiplicity of $V$ in $W$.
For context, in the chapter before this Sagan stated a weaker result that the dimension of Hom$(V,W) = 0$ if and only if there is no submodule of $W$ isomorphic to $V$. This was a corollary of Schur's Lemma.
I don't see how he comes up with this stronger result though. He states it immediately after classifying the endomorphism algebra corresponding to a $G$-module and he says 'similar methods can be used' to obtain Proposition 1.7.10, which implies that they are related but I don't see how.
Can anyone offer some insight into this? At the moment I feel like this proposition just follows from the uniqueness of the decomposition of $W$ into irreducibles via Maschke's Theorem (can someone confirm if I'm correct in thinking this?). But Sagan doesn't prove the uniqueness of the decomposition and seems to be going a different route which I don't follow.
Thanks in advance
Write $W=\bigoplus_{j=1}^m W_j$ with the $W_j$ irreducible. Then $$\text{Hom}(V,W)=\text{Hom}\left(V,\bigoplus_{j=1}^m W_j\right) \cong\bigoplus_{j=1}^m \text{Hom}(V,W_j)$$ and so $$\dim\text{Hom}(V,W)=\sum_{j=1}^m \dim\text{Hom}(V,W_j).$$ By Schur's lemma, $\dim\text{Hom}(V,W_j)=1$ or $0$ according to whether or not $W_j$ is isomorphic to $V$ or not. Therefore $\dim\text{Hom}(V,W)$ is the number of $W_j$ which are isomorphic to $V$.