TITLE: Incarnations of a theorem of Eilenberg
QUESTION [11 upvotes]: 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.

REPLY [4 votes]: There is a paper by Mark Hovey which discusses incarnations of the Eilenberg-Watts theorem in homotopy theory. First he reviews the algebraic version and provides an equivalent formulation wherein the theorem says that the category of abelian groups is left self-contained (note that Eilenberg also proved Ab is right self-contained, though it seems Watts did not prove this). Hovey then finds the correct definition of this concept in homotopy theory and proves a very general theorem about when a model category has this property. His theorem holds for topological spaces, simplicial sets, chain complexes, and all the models of spectra. Note, there is a more recent version of this paper on Hovey's website, but it does not compare the new results to the classical ones as fully.
On page 2 of Hovey's paper above is an open problem which you might be interested in. It asks for conditions such that a closed symmetric monoidal category is self-contained (rather than homotopically self-contained as Hovey's paper answers).