Reflexivity of functors of modules

Let $R$ be an associative ring with unit. %It is natural to We consider $R$-modules as module 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.