I would like to write isomorphisms $A^* \otimes B \cong \hom(A, B)$ and $\hom(A\otimes B, C) \cong \hom(A, B^* \otimes C)$ explicitly.
I think that $\varphi: A^* \otimes B \to \hom(A, B)$ is given by $\varphi(f\otimes b)(a)=f(a)b$. Is the inverse map $\varphi^{-1}: \hom(A, B) \to A^* \otimes B$ given by $\varphi^{-1}(f)=\sum_i f_{i} \otimes b_i$ for some $f_i \in A^*$ and $b_i \in B$ such that $f(a) = \sum_i f_i(a)b_i$?
How to define the maps $\hom(A\otimes B, C) \to \hom(A, B^* \otimes C)$ and $\hom(A, B^* \otimes C) \to \hom(A\otimes B, C)$ explicitly? Thank you very much.