Exercise 4, § 11

Let E be a commutative bigebra over A, $m$ the product $E \otimes E \to E$, $c$ the coproduct $E \to E \otimes E$, $e$ the unit element and $\gamma$ the counit of E. (a) Show that for every commutative A-algebra B, the law of composition which associates with every ordered pair

the A-algebra homomorphism (where $m_B$ is the product in B), defines a monoid structure on $\mathrm{Hom}_{A-\mathrm{alg.}}(E, B)$. (b) In order that, for every commutative A-algebra B, $\mathrm{Hom}_{A-\mathrm{alg.}}(E, B)$ be a group, it is necessary and sufficient that there exist an A-algebra homomorphism $i: E \to E$ such that $i(e) = e$ and the composite mappings $m \circ (1_E \otimes i) \circ c$ and $m \circ (i \otimes 1_E) \circ c$ are both equal to the mapping $x \mapsto \gamma(x)e$. $i$ is then called an inversion in E; it is an isomorphism of E onto the opposite bigebra $E^0$.