Doi-Hopf modules, Yetter-Drinfel'd modules and Frobenius type properties

We study the following question: when is the right adjoint of the forgetful functor from the category of (H, A, C)-Doi-Hopf modules to the category of A-modules also a left adjoint? We can give some necessary and sufficient conditions; one of the equivalent conditions is that C x A and the smash product A # C* are isomorphic as (A, A # C*)-bimodules. The isomorphism can be described using a generalized type of integral. Our results may be applied to some specific cases. In particular, we study the case A = H, and this leads to the notion of k-Frobenius H-module coalgebra. In the special case of Yetter-Drinfel'd modules over a field, the right adjoint is also a left adjoint of the forgetful functor if and only if H is finite dimensional and unimodular.