I'm new to tensor products and I'm trying to figure out how $M^*\otimes M$ is identified with $\text{End}(M)$ in the category of $R$-modules, where $M$ is a free $R$-module of finite rank and $M^*=\text{Hom}(M,R)$.
My attempt is to identify $m^*\otimes m$ with the endomorphism $n\mapsto m^*(m)n$. However, this identification is not bijective because it maps any $m^*\otimes m$ to a diagonal matrix. I am quite confused here. Any help is appreciated.
This is a special case of the more general identification $$ M^* \otimes P \cong \operatorname{Hom}(M, P), $$ which is also a natural isomorphism via the bijection generated by $$ m^* \otimes p \;\longmapsto\; (n \mapsto m^*(n)\,p) $$ for all $n \in M$ and extended linearly. In the special case where $P = M$, this looks like $$ m^* \otimes m \;\longmapsto\; (n \mapsto m^*(n)\,m). $$