An identity related to antipode of a Hopf algebra

Question

Let $H$ be a Hopf algebra with a bijective antipode $S$. Does the equality $\sum\limits_{(h)} h_2 \otimes S^{-1}(h_1) = \sum\limits_{(h)} h_1 \otimes S(h_2)$ hold for any $h \in H$, where $\Delta(h)=\sum h_1 \otimes h_2$? Thank you.

Answer 1

Observe that your requirement is equivalent to the following $$\Delta= (S^2\otimes \mathrm{id})\circ\tau\circ \Delta,$$ where $\tau$ is the flip map.

For a cocommutative Hopf algebra it is known that $S^2=\mathrm{id}$, therefore in that case it is true.

In general, consider the Sweedler's Hopf algebra as an counterexample.