product is $$(M,\sigma_{M, -})\ot (M',\sigma_{M', -})=(M\ot M', \sigma_{M\ot M', -})$$ with $${\label{eq:center2}}
\sigma_{M\ot M', X}=a_{X,M,M'}\circ (\sigma_{M, X}\ot M')\circ a^{-1}_{M,X,M'}
\circ (M\ot \sigma_{M', X})\circ a_{M,M',X},$$ and the unit is $(I,\sigma_{I,-})$, with $${\label{eq:center2bis}}
\sigma_{I,M}=r_M^{-1}\circ l_M.$$ The braiding $c$ on ${\mathcal{Z}}_l({\mathcal{C}})$ is given by $${\label{eq:center3}}
c_{M,M'}=\sigma_{M, M'}:\ (M, \sigma_{M, -})\ot (M', \sigma_{M', -})
\to (M',\sigma_{M', -})\ot (M,\sigma_{M, -}).$$ ${\mathcal{Z}}_l({\mathcal{C}})^{\rm in}$ will be our notation for the monoidal category ${\mathcal{Z}}_l({\mathcal{C}})$, together with the inverse braiding $\tilde{c}$ given by $\tilde{c}_{M,M'}=c^{-1}_{M',M}=\sigma^{-1}_{M', M}$.

The right center ${\mathcal{Z}}_r({\mathcal{C}})$ is defined in a similar way. An object is a couple $(M,\tau_{-, M})$, where $M\in {\mathcal{C}}$ and $\tau_{-, M}:\ -\ot M\to M\ot -$ is a family of natural isomorphisms such that $\tau_{-, I}$ is the natural isomorphism and $${\label{eq:center4}}
a^{-1}_{M, X, Y}\circ \tau_{X\ot Y, M}\circ a^{-1}_{X, Y, M}=(\tau_{X, M}\ot Y)
\circ a^{-1}_{X, M, Y}\circ (X\ot \tau_{Y, M}),$$ for all $X,Y\in {\mathcal{C}}$. A morphism between $(M,\tau_{-, M})$ and $(M',\tau_{-, M'})$ consists of $\vartheta:\ M\to M'$ in ${\mathcal{C}}$ such that $$(\vartheta\ot X)\circ \tau_{X, M}=\tau_{X, M'}\circ (X\ot \vartheta),$$ for all $X\in {\mathcal{C}}$. ${\mathcal{Z}}_r({\mathcal{C}})$ is a braided monoidal category. The unit is $(I, l_{-}^{-1}\circ r_{-})$ and the tensor product is $$(M,\tau_{-, M})\ot (M',\tau_{-, M'})=(M\ot M', \tau_{-, M\ot M'})$$ with $${\label{eq:center5}}
\tau_{X, M\ot M'}=a^{-1}_{M,M',X}\circ (M\ot 