isomorphic to ${}_H^H{\mathcal{YD}}^{\rm in}$.

In a similar way, we can introduce right-right and right-left Yetter-Drinfeld modules. The categories ${\mathcal{YD}}_H^H$ and ${}^H{\mathcal{YD}}_H$ of right-right and right-left Yetter-Drinfeld modules are isomorphic to the right and left center of ${\mathcal{M}}_H$. Let us summarize the results.\
A right-right Yetter-Drinfeld module is a $k$-module $M$ with a right $H$-action and a right $H$-coaction such that $$\begin{aligned}
&&\hspace*{-2cm}\rho(m)= m_{[0]}\ot m_{[1]}\in M\ot_s H;{\label{eq:2.9.1}}\\
&&\hspace*{-2cm} m_{[0]}h_{(1)}\ot m_{[1]}h_{(2)}=(mh_{(2)})_{[0]}\ot 
h_{(1)}(mh_{(2)})_{[1]};{\label{eq:2.9.2}}\end{aligned}$$ or, equivalently, $${\label{eq:2.9.3}}
\rho(mh)=m_{[0]}h_{(2)}\ot S(h_{(1)})m_{[1]}h_{(3)}.$$ The counit condition $m=\varepsilon(m_{[1]})m_{[0]}$ is equivalent to $$m= m_{[0]}\varepsilon(m_{[1]}).$$ The natural isomorphism $\tau_{-,M}$ corresponding to $(M,\rho)\in {\mathcal{YD}}_H^H$ and its inverse are given by the formulas $${\label{eq:2.9.4}}
\tau_{M,V}(v1_{(1)}\ot m1_{(2)})=m_{[0]}\ot mv_{[1]}~~;~~
\tau^{-1}_{M,V}(m1_{(1)}\ot v1_{(2)})=vS^{-1}(m_{[1]})\ot m_{[0]}.$$ Furthermore $$m_{[0]}\varepsilon_t(S^{-2}(m_{[1]}))=m,$$ and $S^{-1}(m_{[1]})\ot m_{[0]}\in H\ot_s M$.\
The monoidal structure on ${\mathcal{YD}}_H^H$ is given by the formula $$\rho(m1_{(1)}\ot n1_{(2)})=m_{[0]}\ot n_{[0]}\ot m_{[1]}n_{[1]}.$$ The braiding is given by [(\[eq:2.9.4\])]{}. The category ${\mathcal{YD}}_H^H$ is isomorphic as a braided monoidal category to ${\mathcal{Z}}_r({\mathcal{M}}_H)$.\
Let $M$ be a right $H$-module and a left $H$-comodule. $M$ is a right-left Yetter-Drinfeld module if one of the three following equivalent conditions is satisfied, for all $m\in M$ and $h\in H$:\
1) $\lambda(m)\in H\ot_sM$ and $$h_{(2)}(mh_{(1)})_{[0]}\ot (mh_{(1)})_{[1]}=
m_{[-1]}h_{(1)}\ot m_{[0]}h_{(2)},$$ 2) $\lambda(mh)=S^{-1}(h_{(3)})m_{[-1]}h_{(1)}\ot m_{[0]}h_{(2)}$;\
3) $(M,\rho)$, with $\rho(m)=m_{[0]}\ot S(m_{[-1]})$ is a right-right Yetter-Drinfeld module.\
