$I$ is an invertible object of ${\mathcal{S}}$, satisfying the following four axioms:

1.  $\phi\Omega={\mathrm{id}}_I\in {\mathrm{End}}_{\mathcal{S}}(I)$. Graphically: $${\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-1.eps}}}} \,=\; {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-1b.eps}}}}\;.$$ Here and in what follows, a black-colored strand is colored by the object $A$, a red-colored strand is colored by the object $I$, and we depict the morphisms $\phi$ and $\Omega$ as $$\phi=\,{\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{phi.eps}}}} {\quad \text{and} \quad}\Omega=\,{\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{Omega.eps}}}}\;\,.$$

2.  Setting $H=(\mu \otimes{\mathrm{id}}_A)({\mathrm{id}}_A \otimes\Delta) {\colon}A \otimes A \to A \otimes A$, we require that $$(\phi \otimes \phi)H=\phi \otimes \phi {\quad \text{and} \quad}H(\Omega \otimes \Omega)=\Omega \otimes \Omega.$$ Graphically: $${\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-2p.eps}}}} \,=\;\, {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-2b.eps}}}} {\quad \text{and} \quad}{\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-2cp.eps}}}} \,=\;\, {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-2d.eps}}}}\;\,.$$

3.  Setting $\nu_{(\phi,\Omega)}={\left\langle\phi S \Omega\right\rangle}_I \in {\mathrm{End}}_{\mathcal{S}}({\mathbb{1}})$, we require that $$\phi S=\nu_{(\phi,\Omega)}\, \phi {\quad \text{and} \quad}S \Omega=\nu_{(\phi,\Omega)}\,\Omega.$$ Graphically: $${\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{gp-3.eps}}}} \,=\,\nu_{(\phi,\Omega)}\;\, {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{gp-3b.eps}}}} {\quad \text{and} \quad}{\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{gp-3c.eps}}}} \,=\,\nu_{(\phi,\Omega)}\;\, {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{gp-3d.eps}}}} \;\,.$$

4.  There is a morphism $f {\colon}A \to A$ in ${\mathcal{S}}$ such that $$\phi \mu \tau_{A,A}=\phi \mu (f \otimes {\mathrm{id}}_A) {\quad \text{and} \quad}\Delta \Omega=(f  \otimes {\mathrm{id}}_A)\Delta \Omega.$$ Graphically: $$\psfrag{h}[Bc][Bc]{\scalebox{.9}{$f$}}
    {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-4.eps}}}} \,=\, {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-4bn.eps}}}} \; {\quad \text{and} \quad}\;
    {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-4p.eps}}}} \,=\, {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-4bpn.eps}}}}\;.$$

5.  There is a morphism $h {\colon}A \to A$ in ${\mathcal{S}}$ such that $$\tau_{A,A} \Delta  \Omega = ({\mathrm{id}}_A \otimes h)\Delta \Omega {\quad \text{and} \quad}\phi \mu=\phi \mu ({\mathrm{id}}_A \otimes h).$$ Graphically: $$\psfrag{h}[Bc][Bc]{\scalebox{.9}{$h$}}
    {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-4c.eps}}}} \,=\, {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-4dn.eps}}}}\; {\quad \text{and} \quad}\;
    {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-4cp.eps}}}} \,=\, {\raisebox{-.4\height}{\scalebox{.9}{\includegraphics{gp-4dpn.eps}}}}\;.$$

In the next 