that the map $\hat G$ is also a derivation with respect to the product $\mu$. And the triple $({\hat{\mathcal F}}, {\hat \Delta}, \mu)$ is an infinitesimal bialgebra. This was pointed out by Marcelo Aguiar.

The dual setup {#s:duality}
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In this section, we present the picture dual to the one in Section \[s:algebra\]. For motivation, we give some simple applications in Section \[ss:sa\]. In Section \[ss:explicit\], we write down explicit formulas for the dual maps. These lead to some immediate consequences, which we discuss in Section \[ss:ssa\].

The product $\centerdot$ and the derivation $S$ {#ss:duality}
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Let ${{\hat{\mathcal F}}^{*}}$ be the restricted dual of ${{\hat{\mathcal F}}}$, namely the space of linear functionals on ${\hat{\mathcal F}}$ that vanish on the graded piece ${{\mathcal F}}_n$ for sufficiently large $n$. As noted before, ${{\hat{\mathcal F}}}$ has a basis consisting of all the [**cd**]{}-monomials. This gives ${{\hat{\mathcal F}}^{*}}$ a natural basis consisting of the delta functions $\delta_v$, where $v$ is any [**cd**]{}-monomial. As vector spaces, ${\hat{\mathcal F}}$ and ${\hat{\mathcal F}}^*$ are isomorphic and we identify them using our specific choice of bases $v \leftrightarrow \delta_v$. This may look unnatural at first, but it is not so strange given that we are indeed biased towards 