Keywords: monoidal Hom-Hopf algebra; separable functors; Maschke type theorem; total integral; relative Hom-Hopf module
Summary: Let $(H,\alpha )$ be a monoidal Hom-Hopf algebra and $(A,\beta )$ a right $(H,\alpha )$-Hom-comodule algebra. We first introduce the notion of a relative Hom-Hopf module and prove that the functor $F $ from the category of relative Hom-Hopf modules to the category of right $(A, \beta )$-Hom-modules has a right adjoint. Furthermore, we prove a Maschke type theorem for the category of relative Hom-Hopf modules. In fact, we give necessary and sufficient conditions for the functor that forgets the $(H, \alpha )$-coaction to be separable. This leads to a generalized notion of integrals.