bijective antipode. Then $S_{|H_t}=({\varepsilon}_s)_{|H_t}$, and $S^{-1}_{|H_s}=(\ol{\varepsilon}_t)_{|H_s}$, so $S$ restricts to an anti-algebra isomorphism $H_t\to H_s$.

It follows that the separability idempotents of $H_t$ and $H_s$ are $e_t=S(1_{(1)})\ot 1_{(2)}$ and $e_s=1_{(1)}\ot S(1_{(2)})$. Consequently, we have the following formulas, for $z\in H_t$ and $y\in H_s$: $$\begin{aligned}
zS(1_{(1)})\ot 1_{(2)}&=&S(1_{(1)})\ot 1_{(2)}z;{\label{eq:1.9.3}}\\
y1_{(1)}\ot 1_{(2)}&=& 1_{(1)}\ot S^{-1}(y)1_{(2)}.{\label{eq:1.9.4}}\end{aligned}$$ Applying $S^{-1}\ot H$ to [(\[eq:1.9.3\])]{}, we find $${\label{eq:1.9.2}}
1_{(1)}S^{-1}(z)\ot 1_{(2)}=1_{(1)}\ot 1_{(2)}z.$$

The center of a monoidal category
---------------------------------

[\[se:1.3\]]{} Let ${\mathcal{C}}=({\mathcal{C}},\ot,I,a,l,r)$ be a monoidal category. The weak left center ${\mathcal{W}}_l({\mathcal{C}})$ is the category with the following objects and morphisms. An object is a couple $(M,\sigma_{M, -})$, with $M\in {\mathcal{C}}$ and $\sigma_{M, -}:\ M\ot -\to -\ot M$ a natural transformation, satisfying the following condition, for all $X,Y\in {\mathcal{C}}$: $${\label{eq:center1}}
(X\ot \sigma_{M, Y})\circ a_{X,M,Y}\circ (\sigma_{M, X}\ot Y)=a_{X,Y,M}\circ \sigma_{M, X\ot Y}
\circ a_{M,X,Y},$$ and such that $\sigma_{M, I}$ is the composition of the natural isomorphisms $M\ot I\cong M\cong I\ot M$. A morphism between $(M,\sigma_{M, -})$ and $(M',\sigma_{M', -})$ consists of $\vartheta:\ M\to M'$ in ${\mathcal{C}}$ such that $$(X\ot \vartheta)\circ \sigma_{M, X}=\sigma_{M', X}\circ (\vartheta\ot X).$$ The left center ${\mathcal{Z}}_l({\mathcal{C}})$ is the full subcategory of ${\mathcal{W}}_l({\mathcal{C}})$ consisting of objects $(M,\sigma_{M, -})$ with $\sigma_{M,-}$ a natural isomorphism. ${\mathcal{Z}}_l({\mathcal{C}})$ is a braided monoidal category. The tensor 