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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 ${\displaystyle F\simeq -\otimes _{R}F(R)}$.

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

References

1. "In what generality does Eilenberg-Watts hold?". MathOverflow.

• 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

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