be a weak Hopf algebra with a bijective antipode. Then $H$ is a left $H^{\rm op}\ot H$-module coalgebra with left action $$(k\ot h)\triangleright c= hck.$$

We easily compute that $$\begin{aligned}
&&\hspace*{-2cm}
\varepsilon((m\ot l)(k_{(2)}\ot h_{(2)}))\varepsilon((k_{(1)}\ot h_{(1)})\triangleright c)\\
&=& \varepsilon(k_{(2)}m)\varepsilon(lh_{(2)})\varepsilon(h_{(1)}ck_{(1)})\\
&=& \varepsilon(lhckm)=\varepsilon(((m\ot l)(k\ot h))\triangleright c).\end{aligned}$$ The other conditions are easily verified.

[\[co:3.3\]]{} Let $H$ be a weak Hopf algebra with bijective antipode. Then we have a weak Doi-Hopf datum $(H^{\rm op}\ot H, H,H)$ and the categories ${}_H{\mathcal{M}}(H^{\rm op}\ot H)^H$ and ${}_H{\mathcal{YD}}^H$ are isomorphic.

The compatibility relation [(\[eq:3.1.2\])]{} reduces to [(\[eq:2.7.1\])]{}.

As we have seen in [Section \[se:1.4\]]{}, weak Doi-Hopf modules are special cases of entwined modules. The entwining map $\psi:\ H\ot H\to H\ot H$ corresponding to the weak Doi-Hopf datum $(H^{\rm op}\ot H, H,H)$ is given by $${\label{eq:3.3.1}}
\psi(h\ot k)=h_{(2)}\ot h_{(3)}kS^{-1}(h_{(1)}).$$

The Drinfeld double
===================

[\[se:4\]]{} Now we consider the particular case where $H$ is finitely generated and projective as a $k$-module, with finite dual basis $\{(h_i,h_i^*)~|~i=1,\cdots,n\}$. Then $H^*$ is also a weak Hopf algebra, in view of the selfduality of the axioms of a weak Hopf algebra. Recall that the comultiplication is given by the formula $\lan \Delta(h^*),h\ot k\ran =\lan h^*,hk\ran$; the counit is evaluation at $1$. Also recall that $H^*$ 