Functors of modules associated with flat and projective modules II

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.