A Duality for Modules over Monoidal Categories of Representations of Semisimple Hopf Algebras

For a finite dimensional semisimple cosemisimple Hopf algebra A and its dual Hopf algebra B, we set up a natural one-to-one correspondence between categories with actions of the monoidal categories of representations of A and of B. This gives a categorical interpretation of the duality for actions of Hopf algebras on algebras.