Abstract. In this paper we observe the categories of paragraded groups and R-modules with respect to the same set of grades ∆, where R is the paragraded ring with the set of grades ∆. We turn our attention to constructing new objects in those categories using the sets of morphisms of grade δ ∈ ∆. This process defines bifunctors which happen to be left exact. Thus we may construct the right derived functor and it turns out that it behaves the same as in the case of the category of abstract modules.