Tensor enriched categorical generalization of the Eilenberg-Watts theorem

Abstract: Let $b$, $b\text{ }!'$ be commutative monoids in a B\'{e}nabou cosmos. Motivated by six-functor formalisms in algebraic geometry, we prove that the category of commutative monoids over $b\otimes b\text{ }!'$ is equivalent to the category of cocontinuous lax monoidal enriched functors between the monoidal enriched categories of right modules over $b$, $b\text{ }!'$.