to the Drinfeld double of [@Bohm; @Nenciu] (see [Proposition \[pr:4.2\]]{}) and anti-isomorphic to the Drinfeld double of [@NVT] (see [Proposition \[pr:4.4\]]{}). In [Section \[se:5\]]{}, we show that the category of finitely generated projective Yetter-Drinfeld modules over a weak Hopf algebra has duality.\
In Sections \[se:1.1\] and \[se:1.2\], we recall some general properties of weak bialgebras and Hopf algebras. Further detail can be found in [@BW; @BohmNSI; @NV]. In [Section \[se:1.3\]]{}, we recall the center construction, and in [Section \[se:1.4\]]{}, we recall the notions of weak Doi-Hopf modules, weak entwining structures and weak smash products.

Preliminary results
===================

[\[se:1\]]{}

Weak bialgebras
---------------

[\[se:1.1\]]{} Let $k$ be a commutative ring. Recall that a weak $k$-bialgebra is a $k$-module with a $k$-algebra structure $(\mu,\eta)$ and a $k$-coalgebra structure $(\Delta,\varepsilon)$ such that $\Delta(hk)=\Delta(h)\Delta(k)$, for all $h,k\in H$, and $$\begin{aligned}
\Delta^2(1)&=& 1_{(1)}\ot 1_{(2)}1_{(1')}\ot 1_{(2')}=
1_{(1)}\ot 1_{(1')}1_{(2)}\ot 1_{(2')},{\label{eq:1.1.1}}\\
\varepsilon(hkl)&=& \varepsilon(hk_{(1)})\varepsilon(k_{(2)}l)=
\varepsilon(hk_{(2)})\varepsilon(k_{(1)}l),{\label{eq:1.1.2}}\end{aligned}$$ for all $h,k,l\in H$. We use the Sweedler-Heyneman notation for the comultiplication, namely $$\Delta(h)=h_{(1)}\ot h_{(2)}=h_{(1')}\ot h_{(2')}.$$ We summarize the elementary properties of weak bialgebras. The proofs are direct applications of the defining axioms (see [@BohmNSI; @NV]). We have idempotent maps $\varepsilon_t,~\varepsilon_s:\ H\to H$ defined by $$\varepsilon_t(h)=\varepsilon(1_{(1)}h)1_{(2)}~~;
\varepsilon_s(h)=1_{(1)}\varepsilon(h1_{(2)}).$$ $\varepsilon_t$ and $\varepsilon_s$ are called the target map 