\tau_{X, M'})\circ a_{M,X,M'}
\circ (\tau_{X, M}\ot M')\circ a^{-1}_{X,M,M'}.$$ The braiding $d$ is given by $${\label{eq:center6}}
d_{M,M'}=\tau_{M, M'}:\ (M, \tau_{-, M})\ot (M', \tau_{-, M'})\to 
(M', \tau_{-, M'})\ot (M, \tau_{-, M}).$$ ${\mathcal{Z}}_r({\mathcal{C}})^{\rm in}$ is the monoidal category ${\mathcal{Z}}_r({\mathcal{C}})$ with the inverse braiding $\tilde{d}$ given by $\tilde{d}_{M,M'}=d^{-1}_{M',M}=\tau^{-1}_{M', M}$.

For details in the case where ${\mathcal{C}}$ is a strict monoidal category, we refer to [@Kassel Theorem XIII.4.2]. The results remain valid in the case of an arbitrary monoidal category, since every monoidal category is equivalent to a strict one. Recall the following result from [@BCP].

[\[pr:1.0.1\]]{} Let ${\mathcal{C}}$ be a monoidal category. Then we have an isomorphism of braided monoidal categories $F:\ {\mathcal{Z}}_l({\mathcal{C}})\to {\mathcal{Z}}_r({\mathcal{C}})^{\rm in}$, given by $$F(M, \sigma_{M, -})=(M, \sigma^{-1}_{M, -})~~{\rm and}~~F(\vartheta)=\vartheta .$$

We have a second monoidal structure on ${\mathcal{C}}$, defined as follows: $$\ol{{\mathcal{C}}}=({\mathcal{C}},\ol{\ot}=\ot\circ\tau,I, \ol{a}, r, l)$$ with $\tau:\ {\mathcal{C}}\times{\mathcal{C}}\to {\mathcal{C}}\times {\mathcal{C}}$, $\tau(M,N)=(N,M)$ and $\ol{a}$ defined by $\ol{a}_{M,N,X}=a^{-1}_{X,N,M}$.

If $c$ is a braiding on ${\mathcal{C}}$, then $\ol{c}$, given by $\ol{c}_{M,N}=
c_{N,M}$ is a braiding on $\ol{{\mathcal{C}}}$. In [@BCP], the following obvious result was stated.

[\[pr:1.0.2\]]{} Let ${\mathcal{C}}$ be a monoidal category. Then $$\ol{{\mathcal{Z}}_l({\mathcal{C}})}\cong {\mathcal{Z}}_r(\ol{{\mathcal{C}}})~~;~~
\ol{{\mathcal{Z}}_r({\mathcal{C}})}\cong {\mathcal{Z}}_l(\ol{{\mathcal{C}}})$$ as braided monoidal categories.

Weak entwining structures and weak smash products
-------------------------------------------------

[\[se:1.4\]]{} The results in 