  Let $R$ be an associative ring with unit. Given an $R$-module $M$, we can
associate the following covariant functor from the category of $R$-algebras to
the category of abelian groups: $S\mapsto M\otimes_R S$. With the corresponding
notion of dual functor, we prove that the natural morphism of functors
$\,\mathcal M\to \mathcal M^{\vee\vee}\,$ is an isomorphism. We prove several
characterizations of the functors associated with flat modules, flat
Mittag-Leffler modules and projective modules.
