$\Delta_n$ as $$\psfrag{E}[cr][cr]{\scalebox{1.5}[2.6]{$\{$}}
\psfrag{H}[cr][cr]{\scalebox{1.5}[2.6]{$\}$}}
\psfrag{n}[Bc][Bc]{\scalebox{.9}{$n$}}
\mu_n=\;{\raisebox{-.6\height}{\scalebox{.9}{\includegraphics{mu-n.eps}}}}
{\quad \text{and} \quad}\Delta_n=\;{\raisebox{-.3\height}{\scalebox{.9}{\includegraphics{Delta-n.eps}}}} \,.$$

Reverse and dual Hopf algebras {#sect-dual-reverse}
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Denote by ${\mathcal{S}}^{\mathrm{rev}}$ the category opposite to ${\mathcal{S}}$ endowed with the opposite monoidal product. Then ${\mathcal{S}}^{\mathrm{rev}}$ is a symmetric monoidal category with the symmetry induced by that of ${\mathcal{S}}$. For example, for any objects $A,B,X,Y \in {\mathrm{Ob}}({\mathcal{S}}^{\mathrm{rev}})={\mathrm{Ob}}({\mathcal{S}})$, $${\mathrm{Hom}}_{{\mathcal{S}}^{\mathrm{rev}}}(A \otimes_{{\mathcal{S}}^{\mathrm{rev}}} \! B, X \otimes_{{\mathcal{S}}^{\mathrm{rev}}} \!  Y)={\mathrm{Hom}}_{{\mathcal{S}}}(Y \otimes X, B \otimes A).$$ Let $A$ be a Hopf algebra in ${\mathcal{S}}$, with product $\mu$, unit $\eta$, coproduct $\Delta$, counit $\varepsilon$, and antipode $S$. Then $A$ becomes a Hopf algebra in ${\mathcal{S}}^{\mathrm{rev}}$ with product $\Delta$, unit $\varepsilon$, coproduct $\mu$, counit $\eta$, and antipode $S$. This Hopf algebra in ${\mathcal{S}}^{\mathrm{rev}}$ is denoted by $A^{\mathrm{rev}}$ and is called the Hopf algebra *reverse* to $A$. If $A$ is involutory, then so is $A^{\mathrm{rev}}$.

Assume that ${\mathcal{S}}$ is left rigid (and so spherical, see Section \[sect-duality\]). The left (co)evaluations induce a dual functor $?^* {\colon}{\mathcal{S}}^{\mathrm{rev}}\to {\mathcal{S}}$ which is a symmetric monoidal equivalence. Thus, if $A$ is a Hopf algebra in ${\mathcal{S}}$, then the image $A^*$ of $A^{\mathrm{rev}}$ under this dual functor is a Hopf algebra in ${\mathcal{S}}$ called the Hopf algebra *dual* to $A$. Explicitly, the product and the coproduct of 