In Hopf algebra texts, it is usually stated that $1=\eta\epsilon\in$Hom($H^C,H^A$) is the identity under convolution.
$\eta$ is the unit, $\epsilon$ is the counit.
My question is, is that a definition, or can it be proved?
Sincere thanks for any help.
(Do let me know if you need any clarification on the above notations.)