= {{\bf c}}v, \:
\mu(v {\otimes}{e}) = v {{\bf c}}$ and $\mu(u {\otimes}v) = u {{\bf d}}v$ for $u,v \in {{\mathcal F}}$. The relation of $\mu$ with the previous two maps is given by Lemma \[l:coderivation\_identity\].

For $v$, a [**cd**]{}-monomial of degree $n$, let $\beta(v)$ be the coefficient of $v$ in $\Psi(B_{n+1})$. In more fancy language, $\beta(v) = \langle \delta_v,\Psi(B_{n+1}) \rangle$. Observe that $\beta({e}) = 1$. We then extend the definition to ${\hat{\mathcal F}}$ by linearity.

Let $v^*$ denote the reverse of the [**cd**]{}-monomial $v$. Also define $(u {\otimes}v)^*$ to be $v^* {\otimes}u^*$.

It is easy to check from the definitions that ${\hat \Delta}(u^*) = ({\hat \Delta}(u))^*$, $\hat G(u^*) = (\hat G(u))^*$ and $\mu((u {\otimes}v)^*) = (\mu(u {\otimes}v))^*$. It is known that $\beta(v) = \beta(v^*)$. This also follows by induction from the second equality and equation .

The main result
---------------

We now state and prove the main result of this section.

\[t:coderivation\] Let ${\hat \Delta}$ and $\hat G$ be as defined before. Then $${\hat \Delta}\circ {\hat G} = ({\operatorname{id}}{\otimes}{\hat G} + {\hat G} {\otimes}{\operatorname{id}})
\circ {\hat \Delta}.$$ In other words, ${\hat G}$ is a coderivation on ${\hat{\mathcal F}}$ with respect to ${\hat \Delta}$.

We evaluate both sides of the 