into $L_{i+1}\oplus L_{i+2}\oplus\dots$. This completes the inductive step, and thus produces the wanted derivation $d$ on $F_{\mathcal Lie}(C[1])$.

In a similar way, we may produce the derivation $g$ of $F_{\mathcal Lie,\,C[1]}C[1]$ over $d$, by decomposing $F_{\mathcal Lie, C[1]} C[1]=L'_1\oplus L'_2\oplus\dots$, where $L'_n$ is given by the space $\left(\bigoplus_{k+l=n-1} \mathcal Lie(n) \otimes C[1]^{\otimes k} \otimes C[1] \otimes C[1]^{\otimes l}\right)_{S_{n}}$. With this notation, $g$ is written as a sum $g=g_1+g_2+\dots$, where $g_i:C[1]\to L'_i$ is lifted to $F_{\mathcal Lie, C[1]} C[1]$ as a derivation over $d$, and $(g_1+\dots+g_i)^2$ only maps into $L'_{i+1}\oplus L'_{i+2}\oplus\dots$.

Using a slight variation of the above method, we may also construct the wanted homotopy $\mathcal Comm$-inner product, i.e. the module map $f$ stated above. More precisely, we build a map $\chi:C[1]\to Mod(F_{\mathcal Lie, C[1]}C^*[1],F_{\mathcal Lie, C[1]}C[1])$, so that $\chi$ is a chain map under the differential $d_1$ on $C[1]$, and the differential $\delta(f)=f\circ h - (-1)^{|f|} g\circ f$ on $Mod(F_{\mathcal Lie, C[1]}C^*[1],F_{\mathcal Lie, C[1]}C[1])$. Since a module map is given by the components $M_n=\bigoplus_{k+l=n-2} \mathcal Lie(n)\otimes C[1]^{\otimes k}\otimes C[1]\otimes C[1]^{\otimes l}\otimes C[1]$, it is enough to construct $\chi$ as a sum $\chi=\chi_2+\chi_3+\dots$, where $\chi_i:C[1]\to M_i$. Now, the lowest component $\chi_2:C[1]\to C[1]\otimes C[1]$ is defined to be the symmetrized Alexander-Whitney comultiplication. For the induction, we assume that $\Upsilon_i:=\chi_2+\dots+\chi_{i-1}$ are local maps such that $D(\Upsilon_i):=\Upsilon_i\circ d_1-\delta\circ \Upsilon_i$ maps only into higher components $M_i\oplus M_{i+1}\oplus\dots$. Let $\epsilon_i:C[1]\to M_i$ be the lowest term of $D(\Upsilon_i)$. Since $D^2=0$ and $\delta$ has $d_1$ as its lowest component, we see that $\epsilon_i$ is $[d_1,.]$-closed, and by the hypothesis of the proposition also locally $[d_1,.]$-exact. These exact te