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.)
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 $f \star \eta\varepsilon = f$.