Let $A$ be a semisimple algebra over $\mathbb{C}$. Given a decomposition $A^{\circ} = \oplus W_i$ of the regular $A$-module $A^{\circ}$ and an irreducible $A$-submodule $M$, I have seen two ways to calculate the number $n_M(A^{\circ})$ of $W_i$ which is isomorphic to $M$. Now I wonder whether there is some connection between these two methods, can anyone please help?
One way is from Issacs' Character Theory of Finite Groups, but also quite common among other textbooks. One decomposes $A^{\circ}$ first into simple algebras $M(A)$ (where $M(A) = \oplus \{W_i: W_i \cong M\}$ denotes the $M$-homogeneous direct summand). For every $x \in A$, we have a natural homomorphism $x_M: M \to M, v \mapsto vx$. This induces an algebra homomorphism $\theta: A \rightarrow \operatorname{End}(M),\, x \mapsto x_M$. Denote its image by $A_M$, one can show $M(A) \cong A_M$. Finally one uses Double Centralizer Theorem to claim that $A_M = \operatorname{End}(M)$, and it follows immediately that $\dim A_M = (\dim M)^2$. Thus $n_M(A^{\circ}) = \frac{\dim A_M}{\dim M} = \dim M$.
The Double Centralizer Theorem is as follows, where $\operatorname{E}_A(M) = \operatorname{Hom}_A(M, M)$ is the centralizer of $A_M$ in $\operatorname{End}(M)$.
Theorem (Double Centralizer): Let $A$ be a semisimple algebra and let $M$ be an irreducible $A$-algebra. Let $D=E_A(M)$. Let $E_D(M)=A_M$.
An alternative way of doing this is from James & Liebeck, Representations and Characters of Groups, which is by counting the dimension of $\operatorname{Hom}_A(A,M)$ in two ways. The first way uses $\dim(\operatorname{Hom}_A(A,M)) = \Sigma_i \dim(\operatorname{Hom}_A(W_i,M))$. Since $\dim(\operatorname{Hom}_A(W_i,M))$ equals $1$ if $W_i \cong M$, and $0$ if $W_i \not\cong M$, one knows $\dim(\operatorname{Hom}_A(A,M)) = n_M(A).$ The second way implies $\dim(\operatorname{Hom}_A(A,M)) = \dim M$. Equalizing the two ways above one gets $n_M(A) = \dim M$.
I think there is some connections between these two ways for the following reasons. Both methods use the space of algebra homomorphisms between modules. Furthermore, the second way constructs a module homomorphism $\phi_v: A^{\circ} \to M, x \mapsto vx, v \in M$, and this homomorphism can also be used in several steps of the first method. I guess there may be some connection between this homomorphism $\phi_v$ and the other homomorphism $\theta$ contructed in the first method, because these two are similar in form, where both are constructed by letting $A$ act on $M$, and because if one views $M$ as a subalgebra of $A$, $x_v$ also becomes an algebra homomorpism. But I am not sure whether there is such connection, and what exactly the connection is. Can anyone please help?