I'm studying the double centralizer theorem, page 19 in this pdf http://math.uchicago.edu/~may/REU2016/REUPapers/Stevens.pdf, but I can't prove an isomorphism.
Consider $A$ an subalgebra semisimple of $\text{End}\ (V)$, $U_{i}$ are the simple modules of $A$ and $W_{i} = \text{Hom}_{A}(U_{i}, V)$
I would like to prove that $$\bigoplus_{i}\text{Hom}_{A}(W_{i} \otimes U_{i}, V) \cong \bigoplus_{i}\text{Hom}(W_{i}, \text{Hom}_{A}(U_{i}, V))$$
Someone can help me, please?
This is just the tensor-hom adjunction:
Given rings $A$ and $B$, modules $X_A$ and $Y_B$, and a bimodule ${}_AM_B$, then we have the isomorphism $$ \mathrm{Hom}_B(X\otimes_AM,Y) \xrightarrow\sim \mathrm{Hom}_A(X,\mathrm{Hom}_B(M,Y)). $$ Explicitly, the map from left to right sends a homomorphism $f$ to $\hat f$, where $\hat f(x)$ is the map $m\mapsto f(x\otimes m)$.