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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.)

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up vote 6 down vote accepted

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

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