We generalize the notion of ends and coends in category theory to the realm of module categories over finite tensor categories. We call this new concept module (co)end. This tool allows us to give different proofs to several known results in the theory of representations of finite tensor categories. As a new application, we present a description of the relative Serre functor for module categories in terms of a module coend, in a analogous way as a Morita invariant description of the Nakayama functor of abelian categories presented in .
Keywords: tensor category, module category
2020 MSC: 18D20, 18M05
Theory and Applications of Categories, Vol. 37, 2021, No. 6, pp 144-188.