In homological algebra, the Eilenberg-Watts theorem states that if $F\colon\text{Mod}_R\to\text{Mod}_S$ is a right-exact coproduct preserving functor of modules, then $F\cong-\otimes_R F(R).$ The proof which you find in Rotman relies on the existence of a generator for $\text{Mod}_R$, the existence free resolutions, and the five lemma, all three things which are particular to the abelian category of modules over a ring.
I'm now wondering whether this theorem will hold for a general category $C$, enriched over $\mathcal{V}.$ The module category of $C$ is the functor category $\hom(C,\mathcal{V})$, which is also the category of presheaves over $C$, which enjoys the nice property of being the free cocompletion of $C$. So a cocontinuous functor out of $\text{Mod}_C$ is completely determined by its values on $C$. Hence the $\mathcal{V}$-category of cocontinuous functors $\text{Mod}_C\to\text{Mod}_D$ is isomorphic to the functors $C\to\text{Mod}_D,$ and by the hom-tensor adjunction, this is functors $C^\text{op}\otimes D\to\mathcal{V}$, also known as bimodules (or spans, correspondences, profunctors, distributors, it has a lot of names).
Perhaps even more easily seen, by the coYoneda lemma, every presheaf is a colimit of representable presheaves, which just explicitly says that so cocontinuous functor $F\cong -\otimes_C F(Y)$, where $Y$ is the Yoneda embedding.
So my question is, is this a correct proof of the Eilenberg-Watts theorem? Are the assumptions of a generator, the existence of free resolutions, and the validity of the five lemma really not necessary for this result?
Best Answer
My advisor Mark Hovey recently updated a paper from 2009 in which he proved the Eilenberg-Watts Theorem in a very general case, namely for model categories. The arxiv version is here. Note that there's a big difference between version 1 and version 2, and in particular version 2 is much shorter. I recommend version 1 for those of a more algebraic dint.
At the Oregon WCATSS13 conference, we also discussed this result for $\infty$-categories. If you want more information, I recommend emailing the organizers, as they have access to the abstracts and notes from most of the talks. In particular, we explicitly related Hovey's results to the results in Schwede and Shipley's paper Stable Model Categories are Categories of Modules. So this seems very related to your second paragraph, especially in situations arising from homotopy theory.
Hope that helps!