I asked this initially in math.stackexchange:

The group algebra $k(G)$ of any group $G$ satisfies as a Hopf algebra the following identities: $$ S\otimes S\circ \Delta=\sigma\circ\Delta\circ S $$ $$ \nabla\circ S\otimes S=S\circ\nabla\circ\sigma $$ where $S$ is the antipode, $\Delta$, the comultiplication, $\nabla$, the multiplication, and $\sigma:x\otimes y\mapsto y\otimes x$.

Is this valid for all Hopf algebras (in any braided monoidal category) or only for some special class?