Adjunctions between Hom and Tensor as endofunctors of (bi-) module category of comodule algebras over a quasi-Hopf algebra.

For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor endofunctors V k - and - kV are left adjoint to some kinds of  Hom-endofunctors of _HM. The units and counits of these adjunctions are formally trivial as in the classical case.The category of (bi-) modules over a quasi-Hopf algebra is monoidal and some generalized versions of  Hom-tensor relations have been stated for these (bi-) module categories. However, the units and counits of adjunctions are not trivial.  For a right comodule algebra A over a quasi-Hopf algebra H, the bimodule category _AM_A need not be monoidal and a tensor endofunctors of this category can not be defined trivially. But the coaction of  H on A induces an action of (bi-) module category of  H on the (bi-) module category of the comodule algebra A.  In this paper, using the action of the monoidal category _HM_H on the bimodule category _AM_A, we introduce suitable versions of tensor and Hom-endofunctors of _AM_A  and generalize varieties the Hom-tensor adjunctions for (bi-) module categories of comodule algebras over a quasi-Hopf algebra H and in any case, we compute the unit and counit of adjunction explicitely.