I am currently reading about module categories and have been not very successful. In this case a module category is a category with an action of a monoidal category. (more information on nLab)
In specific my task is to find a module category $M$ over $G$-vect ($G$-graded vector spaces), s.th. the Functor category $Fun(M, M) \cong G$-rep (representations of $G$), where G is a group.
Unfortunately I don't have any clue how that could look like. It would already be helpful if someone knew any example of a module category! (So that I can get a feeling for what this category could look like) If anyone knows more about this specific example, it would be of course even better!
Thanks in advance!