Tensor Enriched Categorical Generalization of the Eilenberg-Watts Theorem

Abstract

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

The paper was accepted at 16th June 2025, and published at 10th October 2025 in the journal `Applied Categorical Structures'. Some minor typos were corrected during publication process. This is the final version of the paper, and the content is identical to that of the published paper