qa.quantum algebra - Suppose C and D are Morita equivalent fusion categories, can you say anything about R I: C->Z(C)=Z(D)->D?

If C and D are (higher) Morita equivalent fusion categories, then the Drinfel'd centers Z(C) and Z(D) are braided equivalent. Given any fusion category C we have a restriction functor Z(C)->C (by forgetting the "half-braiding"), and adjoint to that an induction functor C->Z(C).

If C and D are Morita equivalent then you can compose the induction and restriction to get a functor C->Z(C)=Z(D)->D. (Actually now that I think about you may need to fix the Morita equivalence in order to actually identify Z(C) and Z(D)?) Is there anything nice one can say about this composition? If C=D then Etingof-Nikshych-Ostrik says that $R circ I(V) = sum_X X otimes V otimes X^*$.

The reason that I ask is that Izumi calculated the induction and restriction graphs for the Drinfel'd center of one of the even parts of the Haagerup subfactor, and I would like to understand the same picture for the other even part.