Hom-tensor relations for (quasi-) comodule algebras

We discuss quasi-Hopf algebras as introduced by Drinfeld and generalize the Hom-tensor adjunctions from the Hopf case to the quasi-Hopf setting, making the module category over a quasi-Hopf algebra H into a biclosed monoidal category. However, in this case, the unit and counit of the adjunction are not trivial and should be suitably modified in terms of the reassociator and the quasi-antipode of the quasi-Hopf algebra H. In a more general case, for a comodule algebra $ \mathcal{B} $ over a quasi-Hopf algebra H, the module category over $ \mathcal{B} $ need not to be monoidal. However, there is an action of a monoidal category on it. Using this action, we consider some kind of tensor and Hom-endofunctors of module category over $ \mathcal{B} $ and generalize some Hom-tensor relations from module category on H to this module category.