1_{(2)}m\equal{{(\ref{eq:1.1.7})}}\varepsilon_t(z1_{(1)})\ot 1_{(2)}m\\
&=&\varepsilon_t(1_{(1)}z)\ot 1_{(2)}m=1_{(1)}\leftact z\ot 1_{(2)}m\\
&&\hspace*{-2cm}
l_M(l_M^{-1}(m))=l_M(\varepsilon_t(1_{(1)})\ot 1_{(2)}m)=m\\
&&\hspace*{-2cm}
r_M^{-1}(r_M(1_{(1)}m\ot 1_{(2)}\leftact z))=r_M^{-1}(\ol{\varepsilon}_s(z)m)=
1_{(1)}\ol{\varepsilon}_s(z)m\ot 1_{(2)}\\
&=& 1_{(1)}m\ot 1_{(2)}z=1_{(1)}m\ot 1_{(2)}\leftact z\\
&&\hspace*{-2cm}
r_M(r_M^{-1}(m))=r_M(1_{(1)}m\ot 1_{(2)})=\ol{\varepsilon}_s(1)m=m.\end{aligned}$$

Weak Hopf algebras
------------------

[\[se:1.2\]]{} A weak Hopf algebra is a weak bialgebra together with a map $S:\ H\to H$, called the antipode, satisfying $${\label{eq:1.3.1}}
S*H=\varepsilon_s,~~H*S=\varepsilon_t,~~{\rm and}~~S*H*S=S,$$ where $*$ is the convolution product. It follows immediately that $${\label{eq:1.3.2}}
S=\varepsilon_s*S=S*\varepsilon_t.$$ If the antipode exists, then it is unique. We will always assume that $S$ is bijective; if $H$ is a finite dimensional weak Hopf algebra over a field, then $S$ is automatically bijective (see [@BohmNSI Theorem 2.10]).

[\[le:1.5\]]{} Let $H$ be a weak Hopf algebra. Then $S$ is an anti-algebra and an anti-coalgebra morphism. For all $h,g\in H$, we have $$\begin{aligned}
\varepsilon_t(hg)&=&
\varepsilon_t(h\varepsilon_t(g))=h_{(1)}\varepsilon_t(g)S(h_{(2)});{\label{eq:1.4.1}}\\
\varepsilon_s(hg)&=&
\varepsilon_s(\varepsilon_s(h)g)=S(g_{(1)})\varepsilon_s(h)g_{(2)};
{\label{eq:1.4.1a}}\\
\Delta(\varepsilon_t(h))&=&h_{(1)}S(h_{(3)})\ot\varepsilon_t(h_{(2)}){\label{eq:1.4.2}}\\
\Delta(\varepsilon_s(h))&=&\varepsilon_s(h_{(2)})\ot S(h_{(1)})h_{(3)}{\label{eq:1.4.2a}}.\end{aligned}$$

[\[le:1.6\]]{} Let $H$ be a weak Hopf algebra. For all $h\in H$, we have $$\begin{aligned}
\varepsilon_t(h)&=&\varepsilon(S(h)1_{(1)})1_{(2)}=
\varepsilon(1_{(2)}h)S(1_{(1)})=S(\ol{\varepsilon}_s(h)){\label{eq:1.6.1}}\\
\varepsilon_s(h)&=&1_{(1)}\varepsilon(1_{(2)}S(h))=
\varepsilon(h1_{(1)})S(1_{(2)})=S(\ol{\varepsilon}_t(h)){\label{eq:1.6.2}}.\end{aligned}$$

[\[co:1.6b\]]{} Let $H$ be a weak Hopf algebra. For all $h\in H$, we have $${\label{eq:1.6.3}}
\varepsilon_t(h_{(1)})\ot h_{(2)}=S(1_{(1)})\ot 1_{(2)}h~~;~~
h_{(1)}\ot \varepsilon_s(h_{(2)})=h1_{(1)}\ot S(1_{(2)}).$$

[\[pr:1.7\]]{} Let $H$ be a weak Hopf algebra. Then $${\label{eq:1.7.1}}
\varepsilon_t\circ S=\varepsilon_t\circ \varepsilon_s= S\circ \varepsilon_s~~;~~
\varepsilon_s\circ S=\varepsilon_s\circ \varepsilon_t= S\circ \varepsilon_t.$$

[\[co:1.8\]]{} Let $H$ be a weak Hopf algebra with 