Let $R$ be any ring, let $\text{Mod}_R$ be the category of right $R$-modules and let $\text{Ab}$ be the category of abelian groups. There is a classical theorem of Eilenberg (I think) which says that for any right exact functor $F:\text{Mod}_R \to \text{Ab}$ which preserves direct sums, there exists a left module structure on $F(R)$ making $F$ naturally isomorphic to the functor $- \otimes_R F(R)$.

Does anyone know any nice "incarnations" of this theorem? By this I mean "nice", simple and concrete right exact functors from $\text{Mod}_R$ to $\text{Ab}$ (for some $R$) preserving direct sums, for which it is not immediately clear that they are given by a tensor product (= isomorphic to a tensor functor) with a left $R$-module, but for which this left $R$-module can still be constructed in a concrete way.