reading Etingof on quantum group

quantum group is just a fancy name for Hopf algebra.

A Hopf algebra is just an algebra with $(\Delta, \epsilon, S)$, coproduct, counit and antipode satisfying a bunch of axioms.

$\epsilon$ and $S$ are determined by $\Delta$. For the algebra of function $O(G)$, the coproduct is interesting. It determines $\epsilon$ to be the restriction to $e$. But then, given a function