Eilenberg-Watts calculus for finite categories and a bimodule Radford $S^4$ theorem

Understand

We obtain Morita invariant versions of Eilenberg-Watts type theorems, relating Deligne products of finite linear categories to categories of left exact as well as of right exact functors.

• This makes it possible to switch between different functor categories as well as Deligne products, which is often very convenient.
• For instance, we can show that applying the equivalence from left exact to right exact functors to the identity functor, regarded as a left exact functor, gives a Nakayama functor.
• The equivalences of categories we exhibit are compatible with the structure of module categories over finite tensor categories.