$s(\tau)=x$. If $s(\tau)=x$, then ${\rm deg}(\tau m)=\tau\sigma\tau^{-1}$.

[\[th:2.4\]]{} Let $H$ be a weak bialgebra. Then the category ${}_H^H{\mathcal{YD}}$ is isomorphic to the weak left center ${\mathcal{W}}_l({}_H{\mathcal{M}})$ of the category of left $H$-modules. If $H$ is a weak Hopf algebra with bijective antipode, then ${}_H^H{\mathcal{YD}}$ is isomorphic to the left center ${\mathcal{Z}}_l({}_H{\mathcal{M}})$

We will restrict to a brief description of the connecting functors; for more detail (in the left-right case), we refer to [@Nenciu Lemma 4.3]. Take $(M,\sigma_{M,-})\in {\mathcal{W}}_l({}_H{\mathcal{M}})$. For each left $H$-module $V$, we have a map $\sigma_{M,V}:\ M\ot_t V\to V\ot_t M$ in ${}_H{\mathcal{M}}$. We will show that the map $$\lambda:\ M\to H\ot_t M,~~\lambda(m)=\sigma_{M,H}(1_{(1)}m\ot 1_{(2)})=
m_{[-1]}\ot m_{[0]}$$ makes $M$ into a Yetter-Drinfeld module. Conversely, let $(M,\lambda)$ is a Yetter-Drinfeld module; a natural transformation $\sigma$ is then defined by the formula $${\label{eq:2.4.1}}
\sigma_{M,V}(1_{(1)}m\ot 1_{(2)}v)=m_{[-1]}v\ot m_{[0]}.$$ Straightforward computations show that $(M,\sigma)\in {\mathcal{W}}_l({}_H{\mathcal{M}})$. If $H$ is a Hopf algebra with invertible antipode, then the inverse of $\sigma_{M,V}$ is $${\label{eq:2.4.2}}
\sigma_{M,V}^{-1}(1_{(1)}v\ot 1_{(2)}m)=m_{[0]}\ot S^{-1}(m_{[-1]})v.$$

From now on, we assume that $H$ is a weak Hopf algebra with bijective antipode. Since the left center of a monoidal category is a braided monoidal category, it follows from [Theorem \[th:2.4\]]{} that ${}_H^H{\mathcal{YD}}$ is a braided monoidal category; a 