The categories of supermodules over different algebras

Given commutative algebra R and R -module U, Hadi et al constructed the category of supermodules CU whose objects are all pairs of R -modules (N, M) where N is direct sum of U and M contains U. If U and W are R-modules then there were relations between the categories CU and CW under some conditions. For instance, if W is direct sum of U then there exists a covariant functor from CU to CW. Further, if U and W are isomorphic then such functor is a category equivalence. In this paper we investigate the relation between two categories of supermodules over different algebras.