is a morphism $S{\colon}A \to A$ in ${\mathcal{S}}$ such that $$\mu(S \otimes {\mathrm{id}}_A)\Delta=\eta \varepsilon=\mu({\mathrm{id}}_A \otimes S)\Delta.$$ If it exists, an antipode is unique and is anti-(co)multipicative, i.e., $$\begin{aligned}
  S \mu& =\mu \tau_{A,A}(S \otimes S), & S \eta&=\eta, \\
\Delta S&=(S \otimes S)\tau_{A,A}\Delta, & \varepsilon S &=\varepsilon.\end{aligned}$$ A *Hopf algebra* in ${\mathcal{S}}$ is a bialgebra in ${\mathcal{S}}$ which admits an invertible antipode. A Hopf algebra $A$ in ${\mathcal{S}}$ is called *involutory* if its antipode $S$ satisfies $S^2={\mathrm{id}}_A$.

Given a Hopf algebra $A$ in ${\mathcal{S}}$, we depict its product $\mu$, unit $\eta$, coproduct $\Delta$, counit $\varepsilon$, and the antipode $S$ as follows: $$\psfrag{A}[Bc][Bc]{\scalebox{.9}{$A$}}
\psfrag{n}[Bc][Bc]{\scalebox{.9}{$n$}}
\mu=\,{\raisebox{-.35\height}{\scalebox{.9}{\includegraphics{mA2.eps}}}}\;, \quad
\eta=\;{\raisebox{-.35\height}{\scalebox{.9}{\includegraphics{uA2.eps}}}}\;, \quad
\Delta=\,{\raisebox{-.35\height}{\scalebox{.9}{\includegraphics{cpA2.eps}}}}\;, \quad
\varepsilon=\;{\raisebox{-.35\height}{\scalebox{.9}{\includegraphics{epsA2.eps}}}}\;, \quad
S=\;{\raisebox{-.35\height}{\scalebox{.9}{\includegraphics{antipode-S2.eps}}}}\;.$$ Here, the underlying color of all the arcs is the object $A$.

For $n \in {\mathbb{N}}$, we define recursively the morphisms $$\label{es-mun-deln}
\mu_n {\colon}A^{\otimes n} \to A {\quad \text{and} \quad}\Delta_n {\colon}A \to A^{\otimes n}$$ as follows: $$\begin{aligned}
&\mu_0=\eta, && \mu_1={\mathrm{id}}_A,  && \mu_{n+1}=\mu(\mu_n \otimes {\mathrm{id}}_A),\\
&\Delta_0=\varepsilon, && \Delta_1={\mathrm{id}}_A, && \Delta_{n+1}=(\Delta_n \otimes{\mathrm{id}}_A)\Delta.\end{aligned}$$ It follows from the (co)associativity and the (co)unitality of the (co)product that for all $n_1, \dots,n_k \in {\mathbb{N}}$, $$\mu_{n_1+\cdots +n_k}=\mu_k(\mu_{n_1} \otimes \cdots \otimes \mu_{n_k}) {\quad \text{and} \quad}\Delta_{n_1+\cdots +n_k}=(\Delta_{n_1} \otimes \cdots \otimes \Delta_{n_k})\Delta_k.$$ We will depict the morphisms $\mu_n$ and 