Exercise 6, § 1

Let E be a bigebra and let $u \in E' = \mathrm{Hom}(E, K)$ (cf. Exercise 5). Let $g \in E$ be such that $\varepsilon(g) = 1$ and $c(g) = g \otimes g$. Show that the mapping $u \mapsto u(g)$ is an algebra homomorphism of $E'$ into K and that the mapping $y \mapsto gy$ is an endomorphism of the cogebra E; show that this endomorphism is an automorphism if E has an inversion.