Understanding the $\operatorname{Hom}_A(T \otimes_B W, V) \cong \operatorname{Hom}_B(W, \operatorname{Hom}_A(T,V))$ isomorphism?

Question

Peter Webb in his book on Representation theory states that if $T$ is an $(A,B)$-module, $V$ is an $A$-module, $W$ is a $B$-module, then $\operatorname{Hom}_A(T \otimes_B W, V) \cong \operatorname{Hom}_B(W, \operatorname{Hom}_A(T,V))$.

He also gives the mutually inverse maps:

$$f\mapsto(w\mapsto(t\mapsto f(t\otimes w)))$$ and $$(t\otimes w \mapsto g(w)(t)) \leftarrow\!\shortmid g$$

How exactly do I check if these are mutually inverse?

Thanks in advance!

Answer 1

You can check that directly. For example, let $g$ be the map $w\mapsto(t\mapsto f(t\otimes w))$. Then what is the map $h:t\otimes w \mapsto g(w)(t)$? Since $$g(w)(t)=f(t\otimes w),$$ $h$ is the same as $f$. The other way is very similar.