the ideal $J$ that is divided out in [@Bohm; @Nenciu] is equal to ${{\rm Ker}\,}p$, and this will imply that $D(H)$ is equal to the Drinfeld double introduced in [@Bohm; @Nenciu]. We first need some Lemmas.

[\[le:4.1\]]{} Let $H$ a weak bialgebra. For all $h^*\in H^*$, $y\in H_s$ and $z\in H_t$, we have $$\begin{aligned}
h^**(y\leftact\varepsilon)&=&
\lan h_{(2)}^*, y\ran  h_{(1)}^*=y\leftact h^*{\label{eq:4.1.1}}\\
h^**(\varepsilon\rightact y)&=&
\lan h_{(1)}^*, y\ran  h_{(2)}^*=h^*\rightact y{\label{eq:4.1.2}}\\
(z\leftact \varepsilon)*h^*&=&
\lan h_{(2)}^*,z\ran h_{(1)}^*=z\leftact h^*{\label{eq:4.1.3}}\\
(\varepsilon\rightact z)*h^*&=&
\lan h_{(1)}^*,z\ran h_{(2)}^*=h^*\rightact z{\label{eq:4.1.4}}\end{aligned}$$

We only prove [(\[eq:4.1.3\])]{}. For all $h\in H$, we have $$\begin{aligned}
&&\hspace*{-2cm}
\lan (z\leftact \varepsilon)*h^*, h\ran=
\lan \varepsilon, h_{(1)}z\ran \lan h^*,h_{(2)}\ran
= \lan \varepsilon, h_{(1)}1_{(1)}z\ran \lan h^*,h_{(2)}1_{(2)}\ran\\
&=& \lan \varepsilon*h^*, hz\ran
= \lan h^*, hz\ran=\lan z\leftact h^*,h\ran = \lan h_{(2)}^*,z\ran
\lan h_{(1)}^*,h\ran.\end{aligned}$$

[\[le:4.1a\]]{} Let $H$ be a weak Hopf algebra with bijective antipode. For all $y\in H_s$, $z\in H_t$, we have $${\label{eq:4.1.5}}
S^{-1}(z)\leftact \varepsilon=z\leftact\varepsilon~~{\rm and}~~
\varepsilon\rightact y=\varepsilon\rightact S^{-1}(y).$$

For all $h\in H$, we have $$\begin{aligned}
&&\hspace*{-2cm}
\lan S^{-1}(z)\leftact \varepsilon,h\ran= \varepsilon(hS^{-1}(z))\equal{{(\ref{eq:1.1.2})}}
\varepsilon(h1_{(1)})\varepsilon(1_{(2)}S^{-1}(z))
\equal{\ref{le:1.5}}
\varepsilon(h1_{(1)})\varepsilon(zS(1_{(2)}))\\
&\equal{{(\ref{eq:1.6.3})}}&
\varepsilon(h_{(1)})\varepsilon(z\varepsilon_s(h_{(2)}))
\equal{{(\ref{eq:1.1.6})}}\varepsilon(\varepsilon_s(h)z)
\equal{{(\ref{eq:1.1.3a})}} \varepsilon(hz)=\lan z\leftact\varepsilon, h\ran.\end{aligned}$$ The second statement can be proved in a similar way.

[\[pr:4.2\]]{} Let $H$ be a finitely generated projective weak Hopf algebra. Then ${{\rm Ker}\,}(p)$ is the $k$-linear span $J$ of elements 