Hopf algebra: Identity under convolution 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.)
 A: It can be proved using the definition of (co-)associativity and the (co-)unit.
Let $\mathbb K$ be a field. Let $(A,m)$ be a associative $\mathbb K$-algebra with unit $\eta: \mathbb K \to A$ and let $(C,\Delta)$ be a coassociative $\mathbb K$-coalgebra with counit $\varepsilon: C \to \mathbb K$.
The convolution $\star: \operatorname{Hom}(C,A) \times \operatorname{Hom}(C,A) \to \operatorname{Hom}(C,A)$ is defined by
$$f \star g := m \circ (f \otimes g) \circ \Delta.$$
Let $\mathbf 1 := \eta(1_{\mathbb K})$, then from the definition of the unit follows
$$m(\mathbf 1 \otimes a) = a \quad \text{for all } a\in A.$$
Furthermore, 
$$(\varepsilon \otimes \operatorname{id}) \circ \Delta = 1_{\mathbb K} \otimes \operatorname{id},$$
by definition of counit. Using this we show
$$\eta \varepsilon \star f = f \star \eta\varepsilon = f \quad \text{for all } f \in\operatorname{Hom}(C,A).$$
For all $c \in C$ we have
\begin{align*}
(\eta\varepsilon \star f)(c) &= (m \circ (\eta \varepsilon \otimes f) \circ \Delta)(c)\\
&=(m \circ (\eta \otimes f) \circ (\varepsilon \otimes \operatorname{id}) \circ \Delta)(c)\\
&= (m \circ (\eta \otimes f))(1_{\mathbb K} \otimes c)\\
&= m(\mathbf 1 \otimes f(c))\\
&= f(c),
\end{align*}
hence $\eta\varepsilon \star f = f$. Similarly one shows $f \star \eta\varepsilon = f$.
