If a group $G$ satisfies $g^2=e, \forall g\in G,$ then $gh=gh(hg)^2=ghhghg=gghg=hg, \forall g, h\in G,$ thus $G$ is commutative.
Since Hopf algebras correspond to groups, one should obtain a similar result for Hopf algebras. Let me adapt the Sweedler's notation here.
The corresponding statement ought to be: if $\epsilon$ denotes the co-unit mapping, $\Delta$ the comultiplication, and if $$\epsilon(a)=\sum a_1a_2, \forall a,$$ then $$\sum a_1\otimes a_2=\sum a_2\otimes a_1.$$ (Here we write $\Delta(a)=\sum a_1\otimes a_2.$)
However, I do not see how we could translate the above group-theoretic proof into one Hopf-algebraic proof. I tried using the associativity, the co-unit property, and the hypothesis, to complete the proof, but to no avail.
Maybe I formulated the statements wrongly, or maybe I missed something important?
It seems that most books treat this as a formal consequence of the group-theoretic result, but I would like to have a proof without using groups.
Thanks in advance, for any hints or answers.