In mathematics, specifically homological algebra, the Eilenberg–Watts theorem tells when a functor between the categories of modules is given by an application of a tensor product. Precisely, it says that a functor F : M o d R → M o d S {\displaystyle F:\mathbf {Mod} _{R}\to \mathbf {Mod} _{S}} is additive, is right-exact and preserves coproducts if and only if it is of the form F ≃ − ⊗ R F ( R ) {\displaystyle F\simeq -\otimes _{R}F(R)} . For a proof, see The theorems of Eilenberg & Watts (Part 1)
References
Charles E. Watts, Intrinsic characterizations of some additive functors, Proc. Amer. Math. Soc. 11, 1960, 5–8. Samuel Eilenberg, Abstract description of some basic functors, J. Indian Math. Soc. (N.S.) 24, 1960, 231–234 (1961).
Further reading Eilenberg-Watts theorem in nLab