In this note, we introduce the (small, pseudo-)ℬ(M,X)-cojective modules and we generalize (small, pseudo-)cojective modules via the class ℬ(M,X). Let M = M₁ ⊕ M₂ be an X-amply supplemented module with the finite internal exchange property. Then for every decomposition of M = M_i ⊕ M_j, M_i is ℬ(M_j,X)-cojective for i ≠ j, M₁ and M₂ are X-lifting if and only if M is X-lifting. We also prove that for an X-amply supplemented module M = M₁ ⊕ M₂ such that M₁ and M₂ are indecomposable X-lifting modules, if M₂ is ℬ(M₁,X)-cojective and M₁ is small-ℬ(M₂,X)-cojective then M is X-lifting.