We find a sufficient condition for the category of entwined Hom-modules to be monoidal. Moreover, we introduce morphisms between the underlying monoidal Hom-algebras and monoidal Hom-coalgebras, which give rise to functors between the category of entwined Hom-modules, and we study tensor identities for monodial categories of entwined Hom-modules. Finally, we give necessary and sufficient conditions for the general induction functor from H (M k)(ψ) C A to H (M k)(ψ) C A to be separable