Opposite algebra induced by a braiding

Let $(\mathcal{C},\otimes ,\mathbf{1},c)$ be a braided monoidal category and let $(A,m,\eta )$ be an algebra object (Separable algebra objects and bimodules in a fusion category). Define the monoidal opposite ${\mathcal{C}}^{\mathrm{m}\mathrm{p}}$ to have the same objects and morphisms, tensor product $X{\otimes }_{\mathrm{m}\mathrm{p}}Y:=Y\otimes X$, associator ${\alpha }_{X,Y,Z}^{\mathrm{m}\mathrm{p}}:={\alpha }_{Z,Y,X}^{-1}$, and unitors ${\lambda }_X^{\mathrm{m}\mathrm{p}}:={\rho }_X$, ${\rho }_X^{\mathrm{m}\mathrm{p}}:={\lambda }_X$. The braiding defines a strong monoidal equivalence $J:{\mathcal{C}}^{\mathrm{m}\mathrm{p}}\to \mathcal{C}$ that is the identity on the underlying category and has tensorator $c_{Y,X}^{-1}:X\otimes Y\to Y\otimes X$.

Regard $A$ as an algebra in ${\mathcal{C}}^{\mathrm{m}\mathrm{p}}$ with the same unit and multiplication $m:A{\otimes }_{\mathrm{m}\mathrm{p}}A=A\otimes A\to A$. Its transport along $J$ is the opposite algebra $A^{\mathrm{o}\mathrm{p}}$ in $\mathcal{C}$; its multiplication is $m_{A^{\mathrm{o}\mathrm{p}}}=m\circ c_{A,A}^{-1}:A\otimes A\to A,$ and its unit is $\eta$. This convention fixes the braiding direction. The hexagon identities are the tensorator coherence of $J$, and its unit coherence follows from the braiding unit equations.