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We obtain Morita invariant versions of Eilenberg-Watts type theorems,relating Deligne products of finite linear categories to categories of leftexact as well as of right exact functors. This makes it possible to switchbetween different functor categories as well as Deligne products, which isoften very convenient. For instance, we can show that applying the equivalencefrom left exact to right exact functors to the identity functor, regarded as aleft exact functor, gives a Nakayama functor. The equivalences of categories weexhibit are compatible with the structure of module categories over finitetensor categories. This leads to a generalization of Radford's $S^4$-theorem tobimodule categories. We also explain the relation of our construction torelative Serre functors on module categories that are constructed via inner Homfunctors.

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