over a weak bialgebra $H$, and show that it is isomorphic to the category of Yetter-Drinfeld modules. If $H$ is a weak Hopf algebra, then the weak center equals the center. In this situation, properties of the center construction can be applied to show that the four categories of Yetter-Drinfeld modules, namely the left-left, left-right, right-left and right-right versions, are isomorphic as braided monoidal categories. Here we apply methods that have been used before in [@BCP], in the case of quasi-Hopf algebras.\
In [@cmz], it was observed that Yetter-Drinfeld modules over a classical Hopf algebra are special cases of Doi-Hopf modules, as introduced by Doi and Koppinen (see [@Doi92; @Koppinen95]). In [Section \[se:3\]]{}, we will show that Yetter-Drinfeld modules over weak Hopf algebras are weak Doi-Hopf modules, in the sense of Böhm [@Bohm], and, a fortiori, weak entwined modules [@CDG], and comodules over a coring [@BW].\
The advantage of this approach is that it leads easily to a new description of the Drinfeld double of a finitely generated projective weak Hopf algebra, using methods developed in [@CDG]: we define the Drinfeld double as a weak smash product of $H$ and its dual. We show that our Drinfeld double is equal 