Reflexivity of modules

Abstract: We consider $\,R-$modules as functors in the following way: if $\,M\,$ is a (left) $R$-module, let $\,\mathcal M\,$ be the functor of $\,\mathcal R-$modules defined by $\,\mathcal M(S) := S \otimes_R M\,$ for every $\,R-$algebra $\,S$. With the corresponding notion of dual functor, we prove that the natural morphism of functors $\,\mathcal M\to \mathcal M^{**}\,$ is an isomorphism.