For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor functors introduced by V are known to be left adjoint to some kind of Homfunctors as endofunctors of H. The units and counits of adjunctions, in this case, are formally trivial as in the classical case. In this paper, we generalize this Hom-tensor adjunction for (bi-)module categories over a quasi-Hopf algebra H and show that these (bi-)module categories are biclosed monoidal. However, the units and counits of adjunctions in these generalized cases are not as trivial as in the Hopf algebra case, and they should be modified in terms of the reassociator and the quasi-antipode. Also, if the H-module V is finitely generated and projective as a k-module, we will obtain a generalized form of adjunction between the tensor functors −⊗V and −⊗V∗ depending on the reassociator and quasi-antipode of H and describe a natural isomorphism between functors −⊗V∗ and some kinds of Hom-functor explicitly. Furthermore, we consider the special case V = A being an Hmodule algebra. In this case, each tensor functor will be a monad and its corresponding right adjoint is a comonad. We describe isomorphisms between the (Eilenberg–Moore) module categories over these monads and the (Eilenberg–Moore) comodule categories over their corresponding comonads explicitly.
