the elements of the center of the ${\mathcal{C}}$-bimodule category ${{^{\#^{-1}}_H\mathcal{M}}}$ are nothing but the “duals" of the well-known anti Yetter-Drinfeld modules over $H$. On the other hand, if we pursue a contravariant theory, then the bimodule category of interest is ${{^{\#}_H\mathcal{M}}}$ and we show that $(F,M)$ is a stable central contrapair if $$F:=\Hom_H(-,1),$$ $M$ is the usual anti Yetter Drinfeld module and the stability condition is the usual one [@hkrs1].

We recall that the center of a monoidal category has been studied for different reasons. It is known that the elements of the center of the monoidal category of modules over a Hopf algebra, a weak Hopf algebra, and a quasi Hopf algebra are in fact the Yetter-Drinfeld modules which are the solutions of the quantum Yang-Baxter equations. If we consider this monoidal category as a bimodule category over itself by the left and right actions given by the monoidal tensor product, then the Yetter-Drinfeld modules do indeed form the center of this bimodule category.

One notes that the language of monoidal categories is fundamental to the study of Hopf-like objects. More precisely, for such an object, the category of left modules over it is a monoidal category. Often the 