a_n) = \\[0.3cm]
\displaystyle
\sum_{p =1}^{k}\sum_{{{\lambda}}\in {{\rm Sh}}_{p,k-p}}
(-1)^{{{\varepsilon}}_{{{\lambda}}}}(Q^{{{\mathfrak{c}}}} (v_{{{\lambda}}(1)}, \dots, v_{{{\lambda}}(p)}), v_{{{\lambda}}(p+1)},
\dots, v_{{{\lambda}}(k)} ; a_1, \dots, a_n ) +
\end{array}$$ $$\sum_{1\le t \le s \le n}^{0 \le p \le k}
\sum_{{{\lambda}}\in {{\rm Sh}}_{p,k-p}}
(-1)^{{{\varepsilon}}_{{{\lambda}}} + {{\varepsilon}}^{{{\lambda}}}_{t,p}}(v_{{{\lambda}}(1)}, \dots, v_{{{\lambda}}(p)} ;
a_1, \dots, Q^{{{\mathfrak{o}}}}(v_{{{\lambda}}(p+1)},
\dots, v_{{{\lambda}}(k)}; a_{t}, \dots, a_{s} ) \dots, a_n ) +$$ $$\sum_{1 \le t \le n+1}^{0 \le p \le k-1}
\sum_{{{\lambda}}\in {{\rm Sh}}_{p,k-p}}
(-1)^{{{\varepsilon}}_{{{\lambda}}} + {{\varepsilon}}^{{\lambda}}_{t,p}}(v_{{{\lambda}}(1)}, \dots, v_{{{\lambda}}(p)}, ;
a_1, \dots, a_{t-1}, Q^{{{\mathfrak{o}}}}(v_{{{\lambda}}(p+1)},
\dots, v_{{{\lambda}}(k)}), a_{t}, \dots, a_n )$$ where $$\label{ve-la}
{{\varepsilon}}_{{{\lambda}}} = \sum_{i<j ~ {{\lambda}}(i) > {{\lambda}}(j)}
|v_i||v_j |\,,$$ and $$\label{ve-la-tp}
{{\varepsilon}}^{{{\lambda}}}_{t,p} = \sum_{j=1}^{p} |v_{{{\lambda}}(j)}| +
\sum_{j=p+1}^{k} \sum_{l=1}^{t-1} |v_{{{\lambda}}(j)}| (|a_l|+1) +
\sum_{l=1}^{t-1} (|a_l|+1)\,.$$

Restricting the codifferential ${\hat{Q}}^{{{\mathfrak{c}}}}$ to the coalgebra $$\label{cocomm-V}
{{\Lambda}}^2{{\bf cocomm}}(V) \subset {{\bf Ger}}^{\vee}(V)$$ we get an ${{\Lambda}}{{\bf Lie}}_{\infty}$-algebra structure on $V$.

An $L_{\infty}$-morphism from $V$ to ${C^{\bullet}}(A,A)$ is a degree $0$ map $$\label{U}
U : {{\Lambda}}^2 {{\bf cocomm}}(V) \to {C^{\bullet}}(A,A)$$ satisfying the coherence equations $$\label{UandQ}
\begin{array}{c}
\displaystyle
\sum_{p=1}^k \sum_{{{\lambda}}\in {{\rm Sh}}_{p,k-p}}
(-1)^{{{\varepsilon}}_{{{\lambda}}}} U(Q^{{{\mathfrak{c}}}} (v_{{{\lambda}}(1)}, \dots, v_{{{\lambda}}(p)}), v_{{{\lambda}}(p+1)},
\dots, v_{{{\lambda}}(k)})\\[0.3cm]
\displaystyle
= -[m , U (v_1, \dots, v_k)]_G \, - \\[0.3cm]
\displaystyle
\sum_{p=1}^{k-1} \sum_{{{\lambda}}\in {{\rm Sh}}_{p,k-p}}
(-1)^{{{\varepsilon}}_{{{\lambda}}} + |v_{{{\lambda}}(1)}| + \dots + |v_{{{\lambda}}(p)}|}
[U(v_{{{\lambda}}(1)}, \dots, v_{{{\lambda}}(p)}) ,
U(v_{{{\lambda}}(p+1)}, \dots, v_{{{\lambda}}(k)}) ]_G\,,
\end{array}$$ where ${{\varepsilon}}_{{{\lambda}}}$ 