On the tensor product of modules over skew monoidal actegories

This paper is about skew monoidal tensored VV-categories (= skew monoidal hommed VV-actegories) and their categories of modules. A module over 〈M,⁎,R〉〈M,⁎,R〉 is an algebra for the monad T=R⁎ _ on MM. We study in detail the skew monoidal structure of MTMT and construct a skew monoidal forgetful functor MT→ME to the category of E  -objects in MM where E=M(R,R)E=M(R,R) is the endomorphism monoid of the unit object R  . Then we give conditions for the forgetful functor to be strong monoidal and for the category MTMT of modules to be monoidal. In formulating these conditions a notion of ‘self-cocomplete’ subcategories of presheaves appears to be useful which provides also some insight into the problem of monoidality of the skew monoidal structures found by Altenkirch, Chapman and Uustalu on functor categories [C,M][C,M].