The reverse and the opposite of a monoidal category

Let $(\mathcal{C},\otimes ,\mathbf{1},\alpha ,\lambda ,\rho )$ be a monoidal category (Monoidal category).

Its reverse monoidal category ${\mathcal{C}}^{\mathrm{r}\mathrm{e}\mathrm{v}}$ has the same underlying category and unit object, but tensor

$X{\otimes }^{\mathrm{r}\mathrm{e}\mathrm{v}}Y:=Y\otimes X\mathrm{.}$

On morphisms, $f:X\to X^{\mathrm{\prime }}$ and $g:Y\to Y^{\mathrm{\prime }}$ are sent to $f{\otimes }^{\mathrm{r}\mathrm{e}\mathrm{v}}g:=g\otimes f:Y\otimes X⟶Y^{\mathrm{\prime }}\otimes X^{\mathrm{\prime }}\mathrm{.}$

The associator of ${\mathcal{C}}^{\mathrm{r}\mathrm{e}\mathrm{v}}$ is

${\alpha }_{X,Y,Z}^{\mathrm{r}\mathrm{e}\mathrm{v}}:((X{\otimes }^{\mathrm{r}\mathrm{e}\mathrm{v}}Y){\otimes }^{\mathrm{r}\mathrm{e}\mathrm{v}}Z)=Z\otimes (Y\otimes X)\stackrel{{\alpha }_{Z,Y,X}^{-1}}{\to }(Z\otimes Y)\otimes X=X{\otimes }^{\mathrm{r}\mathrm{e}\mathrm{v}}(Y{\otimes }^{\mathrm{r}\mathrm{e}\mathrm{v}}Z),$

and its unit constraints are

${\lambda }_X^{\mathrm{r}\mathrm{e}\mathrm{v}}:={\rho }_X,{\rho }_X^{\mathrm{r}\mathrm{e}\mathrm{v}}:={\lambda }_X\mathrm{.}$

Its opposite monoidal category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ is the opposite category (Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$) with the same objects, the same unit object, and tensor bifunctor defined on morphisms by $f^{\mathrm{o}\mathrm{p}}\otimes g^{\mathrm{o}\mathrm{p}}:=(f\otimes g{)}^{\mathrm{o}\mathrm{p}}\mathrm{.}$ Its structure maps are obtained from the inverse original isomorphisms:

${\alpha }_{X,Y,Z}^{\mathrm{o}\mathrm{p}}:=(({\alpha }_{X,Y,Z}^{-1}{)}^{\mathrm{o}\mathrm{p}}):((X\otimes Y)\otimes Z)\to X\otimes (Y\otimes Z),$

${\lambda }_X^{\mathrm{o}\mathrm{p}}:=(({\lambda }_X^{-1}{)}^{\mathrm{o}\mathrm{p}}):(\mathbf{1}\otimes X)\to X,{\rho }_X^{\mathrm{o}\mathrm{p}}:=(({\rho }_X^{-1}{)}^{\mathrm{o}\mathrm{p}}):(X\otimes \mathbf{1})\to X,$

where passage to ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ reverses the direction of the original inverse isomorphisms.

These two constructions are different: reversing the tensor order is not the same operation as reversing all morphisms.