Eilenberg–Watts theorem

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 ${\displaystyle F:\mathbf {Mod} _{R}\to \mathbf {Mod} _{S}}$ is additive, is right-exact and preserves coproducts if and only if it is of the form ${\displaystyle F\simeq -\otimes _{R}F(R)}$.^[1]

For a proof, see The theorems of Eilenberg & Watts (Part 1)