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.



Incarnations of a theorem of Eilenberg
More generally, the Theorem of Eilenberg-Watts says the following: The category of cocontinuous functors $\mathrm{Mod}(R) \to \mathrm{Mod}(S)$ is equivalent to the category of $(R,S)$-bimodules. A bimodule ${}\_R M \_{S}$ corresponds to the functor $- \otimes\_R M$. There is a recent paper by A. Nyman which deals more generally with cocontinuous functors $\mathrm{Qcoh}(X) \to \mathrm{Qcoh}(Y)$ for nice schemes $X,Y$.



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> *The Eilenberg-Watts theorem over schemes*, J. Pure Appl. Algebra, 214 (2010), 1922-1954; [online](http://myweb.facstaff.wwu.edu/nymana/ewschemes.pdf).
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As for your question, take a finitely generated projective module $P$. Then $\mathrm{Hom}(P,-) : \mathrm{Mod}(R) \to \mathrm{Mod}(R)$ is right exact (since $P$ is projective) and preserves infinite direct sums (since $P$ is finitely generated), thus cocontinuous. In this case the Theorem shows that $\mathrm{Hom}(P,-) \cong (-) \otimes P^\*$, where $P^\* = \mathrm{Hom}(P,R)$ is the dual of $P$. More generally, for every vector bundle $V$ on some scheme/manifold we have $\underline{\mathrm{Hom}}(V,-) \cong (-) \otimes V^\*$.


