map. Then $${\label{eq:2.1.3}}
\varepsilon(m_{[-1]})m_{[0]}=\varepsilon_t(m_{[-1]})m_{[0]}.$$ Consequently, in the definition of a Yetter-Drinfeld module, the counit property $\varepsilon(m_{[-1]})m_{[0]}=m$ can be replaced by $\varepsilon_t(m_{[-1]})m_{[0]}=m$.

$$\varepsilon_t(m_{[-1]})m_{[0]}=\varepsilon(1_{(1)}m_{[-1]})1_{(2)}m_{[0]}=
\varepsilon(m_{[-1]})m_{[0]}.$$

In the case of a weak Hopf algebra, the compatibility relation [(\[eq:2.1.2\])]{} can also be restated:

[\[pr:2.2\]]{} (cf. [@Nenciu Remark 2.6]) Let $H$ be a weak Hopf algebra, and $M$ a $k$-module, with a left $H$-action and a left $H$-coaction. $M$ is a Yetter-Drinfeld module if and only if $${\label{eq:2.2.1}}
\lambda(hm)=h_{(1)}m_{[-1]}S(h_{(3)})\ot h_{(2)}m_{[0]}.$$

Let $M$ be a Yetter-Drinfeld module. Then we compute $$\begin{aligned}
&&\hspace*{-15mm}
h_{(1)}m_{[-1]}S(h_{(3)})\ot h_{(2)}m_{[0]}=
(h_{(1)}m)_{[-1]}h_{(2)}S(h_{(3)})\ot  (h_{(1)}m)_{[0]}\\
&=& (h_{(1)}m)_{[-1]}\varepsilon_t(h_{(2)})\ot  (h_{(1)}m)_{[0]}
\equal{{(\ref{eq:1.1.3})}}
(1_{(1)}hm)_{[-1]}1_{(2)}\ot  (1_{(1)}hm)_{[0]}\\
&\equal{{(\ref{eq:2.1.2})}}& 1_{(1)}(hm)_{[-1]}\ot
1_{(2)}(hm)_{[0]}\equal{{(\ref{eq:2.1.1})}}
(hm)_{[-1]}\ot (hm)_{[0]}=\lambda(hm).\end{aligned}$$ Conversely, assume that [(\[eq:2.2.1\])]{} holds for all $h\in H$ and $m\in M$. Taking $h=1$ in [(\[eq:2.2.1\])]{}, we find $$\begin{aligned}
\lambda(m)&=&1_{(1)}m_{[-1]}S(1_{(3)})\ot 1_{(2)}m_{[0]}\\
&=& 1_{(1)}m_{[-1]}S(1_{(2')})\ot 1_{(2)}1_{(1')}m_{[0]}\in H\ot_t M\end{aligned}$$ and $$\begin{aligned}
\lambda(m)&=& 1_{(1)}m_{[-1]}S(1_{(3)})\ot 1_{(2)}m_{[0]}
= 1_{(1)}m_{[-1]}S(1_{(2')})\ot 1_{(1')}1_{(2)}m_{[0]}\nonumber\\
&=& m_{[-1]}S(1_{(2')})\ot 1_{(1')}m_{[0]}.{\label{eq:2.2.2}}\end{aligned}$$ Now $$\begin{aligned}
&&\hspace*{-15mm}
(h_{(1)}m)_{[-1]}h_{(2)}\ot (h_{(1)}m)_{[0]}\equal{{(\ref{eq:2.2.1})}}
h_{(1)}m_{[-1]}S(h_{(3)})h_{(4)}\ot h_{(2)}m_{[0]}\\
&=& h_{(1)}m_{[-1]}\varepsilon_s(h_{(3)})\ot h_{(2)}m_{[0]}
\equal{{(\ref{eq:1.6.3})}}h_{(1)}m_{[-1]}S(1_{(2)})\ot h_{(2)}1_{(1)}m_{[0]}\\
&\equal{{(\ref{eq:2.2.2})}}&
h_{(1)}m_{[-1]}\ot h_{(2)}m_{[0]},\end{aligned}$$ as needed.

[\[co:2.2bis\]]{} Let $M$ be a left-left Yetter-Drinfeld module. For all $y\in H_s$, $z\in H_t$ and $m\in M$, we have $${\label{eq:2.2bis.1}}
\lambda(zm)=zm_{[-1]}\ot m_{[0]}~~;~~\lambda(ym)=m_{[-1]}S(y)\ot m_{[0]}.$$

$$\begin{aligned}
\lambda(zm)&\equal{(\ref{eq:1.1.4},\ref{eq:2.2.1})}&
1_{(1)}zm_{[-1]}S(1_{(3)})\ot 1_{(2)}m_{[0]}\\
&\equal{{(\ref{eq:1.1.6})}}&
z1_{(1)}m_{[-1]}S(1_{(3)})\ot 1_{(2)}m_{[0]}
\equal{{(\ref{eq:2.2.1})}} zm_{[-1]}\ot m_{[0]}.\end{aligned}$$

The other assertion is proved in a similar way.

[\[co:2.3\]]{} Let $M$ 