be a left-left Yetter-Drinfeld module over a weak Hopf algebra with bijective antipode. Then we have the following identities, for all $m\in M$: $${\label{eq:2.3.1}}
1_{(1)}m_{[0]}\ot 1_{(2)}S^{-1}(m_{[-1]})= m_{[0]}\ot S^{-1}(m_{[-1]});$$ $${\label{eq:2.3.2}}
\varepsilon_s(S^{-2}(m_{[-1]}))m_{[0]}=m.$$

Apply $S^{-1}$ to the first factor of [(\[eq:2.2.2\])]{}, and then switch the two tensor factors. Then we obtain [(\[eq:2.3.1\])]{}. [(\[eq:2.3.2\])]{} is proved as follows: $$\begin{aligned}
&&\hspace*{-2cm}
m=\varepsilon_t(m_{[-1]})m_{[0]}\equal{{(\ref{eq:2.1.3})}}
\varepsilon(m_{[-1]})m_{[0]}
= \varepsilon(S^{-1}(m_{[-1]}))m_{[0]}\\
&\equal{{(\ref{eq:2.3.1})}}&
\varepsilon(1_{(2)}S^{-1}(m_{[-1]}))1_{(1)}m_{[0]}
\equal{{(\ref{eq:1.6.2})}}\varepsilon_s(S^{-2}(m_{[-1]}))m_{[0]}.\end{aligned}$$

The category of left-left Yetter-Drinfeld modules and left $H$-linear, left $H$-colinear maps will be denoted by ${}_H^H{\mathcal{YD}}$.

[\[ex:2.3\]]{} Let $G$ be a groupoid, and $kG$ the corresponding groupoid algebra. Then $kG$ is a weak Hopf algebra. Let $M$ be a left-left Yetter-Drinfeld module. Then $M$ is a $kG$-comodule, so $M$ is graded by the set $G$, that is $$M=\bigoplus_{\sigma\in G_1} M_\sigma,$$ and $\lambda(m)=\sigma\ot m$ if and only if $m\in M_\sigma$, or ${\rm deg}(m)=\sigma$.\
Recall that the unit element of $kG$ is $1=\sum_{x\in G_0} x$, where $x$ is the identity morphism of the object $x\in G_0$. Take $m\in M_\sigma$. Using [(\[eq:2.2.1\])]{}, we find $$\lambda(m)=\lambda(1m)=\sum_{x\in G} x\sigma x\ot xm=0,$$ unless $s(\sigma)=\tau(\sigma)=x$. So we have $$M= \bigoplus_{\sigma\in G_1\atop s(\sigma)=t(\sigma)} M_\sigma.$$ Take $m\in M_\sigma$, with $s(\sigma)=\tau(\sigma)$, and $\tau\in G_1$. It follows from [(\[eq:2.2.1\])]{} that $\lambda(\tau m)= \tau\sigma\tau^{-1}\ot \tau m =0$, unless 