This paper deals with generalized local homology and generalized local cohomology modules belong to a Serre category of the category of R-modules under some conditions. For an ideal I of R, the concept of the condition CI on a Serre category which is dual to the condition CI of Melkersson is defined. As a main result, it is shown that for a finitely generated R-module M with pd(M) < ∞ and a minimax R-module N of any Serre category S satisfying the condition CI , the generalized local homology HIi (M,N) belongs to S for all i > pd(M). Also, if S satisfies the condition CI , then the generalized local cohomology module Hi I (M,N) ∈ S for all i > pd(M
