In the study of the structure of graded algebras (such as graded ideals,graded subspaces,and radicals) or graded polynomial identities,the grading group can be replaced by any other group that realizes the same grading.Here we come to the notion of weak equivalence of gradings:two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component.Each group grading on an algebra can be weakly equivalent to G-gradings for many different groups G;however,it tus out that there is one distinguished group among them,called the universal group of the grading.In this paper we study categories and functors related to the notion of weak equivalence of gradings.In particular,we introduce an oplax 2-functor that assigns to each grading its support,and show that the universal grading group functor has neither left nor right adjoint.