Identification of semisimple representations I'm reading the book Introduction to Representation Theory by Pavel Etingof et all, and I have some questions about the remark $3.1.3$. The authors state that by Schur's lemma, any semisimple representation $V$ of $A$ is caninically identified with
$$\bigoplus_{X}Hom_{A}(X, V) \otimes X$$
where $X$ runs over all irreducible representations of A.
They claim that the function $f: \bigoplus_{X}Hom_{A}(X, V) \otimes X \to V$ given by $g \otimes x \mapsto g(x)$, $x \in X$, $g \in Hom_{A}(X, V)$ is an isomorphism.
My first question is where is the Schur's lemma being used? And how can i prove that this function is injective?
Thank you in advance
 A: All vector spaces are over some fixed field $k$, that is often omitted. Let us now write (by definition, non-canonically) $V$ as a direct sum of irreducible pieces, $V=Y_1\oplus Y_2\oplus\dots$ and use this below
$$
\begin{aligned}
&\bigoplus_{X}Hom_{A}(X, V) \otimes X
\\
&=
\bigoplus_{X}Hom_{A}(X,Y_1\oplus Y_2\oplus\dots ) \otimes X
\\
&=
\bigoplus_{X}\Big(\ Hom_{A}(X,Y_1)\oplus Hom_{A}(X,Y_2)\oplus\dots\ \Big) \otimes X
\\
&=
\bigoplus_{X}\Big(\ Hom_{A}(X,Y_1)\otimes X\oplus Hom_{A}(X,Y_2)\otimes X\oplus\dots\ \Big) 
\\
&=
\bigoplus_{X}Hom_{A}(X,Y_1)\otimes X\ \oplus\ \bigoplus_{X} Hom_{A}(X,Y_2)\otimes X\oplus\dots
\\
&=
\bigoplus_{X\ :\ X=Y_1}Hom_{A}(X,Y_1)\otimes X
\ \oplus\ 
\bigoplus_{X\ :\ X=Y_2} Hom_{A}(X,Y_2)\otimes X\ \oplus\ \dots
\text{ (Schur)}
\\
&=
Hom_{A}(Y_1,Y_1)\otimes Y_1
\ \oplus\ 
Hom_{A}(Y_2,Y_2)\otimes Y_2\ \oplus\ \dots
\\
&=
k\otimes Y_1
\ \oplus\ 
k\otimes Y_2\ \oplus\ \dots
\text{ (Schur)}
\\
&=Y_1\oplus Y_2\oplus\dots
\\
&=V\ .
\end{aligned}
$$
In the chain of isomorphisms above (denoted by abuse with $=$) only the first and the last step are non-canonical. So, to see that the declared map (induced on each summand by) $g\otimes x\to g(x)$ is canonical, we have only to show that for any
$$
\text{choice }V\overset f\longrightarrow \underbrace{Y_1\oplus Y_2\oplus\dots}_W
$$
we have the commutativity
$\require{AMScd}$
\begin{CD}
\bigoplus_{X}Hom_{A}(X, V) \otimes X @>>> 
\bigoplus_{X}Hom_{A}(X, W) \otimes X
\\
@V V V @VV  V\\
    V @>>f> W
\end{CD}
thus equivalently for every piece:
\begin{CD}
Hom_{A}(X, V) \otimes X @>>> 
Hom_{A}(X, W) \otimes X
\\
@V V V @VV  V\\
    V @>>f> W
\end{CD}
and on elements, the diagram chasing is:
\begin{CD}
g \otimes x @>>> 
f_*(g) \otimes x
\\
@V V V @VV  V\\
    g(x) @>>f> f(g(x))
\end{CD}
which is true.
A: Schur's lemma says, some map under some conditions is an isomorphism.  
Your question is to see the obvious map  $\bigoplus_X (\text{Hom}_A(X,V)\otimes X)\rightarrow V$ to be an isomorphism. 
Suppose for simplicity $V$ is an irreducible representation of $A$. 
Hint : How many $A$-linear maps can you think of from $X$ to $V$ if $X$ and $V$ are not isomorphic?
As $V$ is semisimple, it is direct sum of irreducible representations. Suppose $V=V_1\oplus V_2$. 
Check : Does the following equality holds?
 $$\bigoplus_X (\text{Hom}_A(X,V)\otimes X)=\bigg(
\bigoplus_X (\text{Hom}_A(X,V_1)\otimes X)\bigg)\bigoplus \bigg(
\bigoplus_X (\text{Hom}_A(X,V_2)\otimes X)\bigg)$$
Now, $X$ and $V_1,V_2$ are irreducible representations. Can you see the result?
