Tensor Enriched Categorical Generalization of the Eilenberg-Watts Theorem

Let $b$, $b^{\mathrm{\prime }}$ be commutative monoids in a Bénabou cosmos. Motivated by six-functor formalisms in algebraic geometry, we prove that the category of commutative monoids over $b\otimes b^{\mathrm{\prime }}$ is equivalent to the category of cocontinuous lax monoidal enriched functors between the monoidal enriched categories of right modules over $b$, $b^{\mathrm{\prime }}$.