is defined in (\[ve-la\]), $[\,,\,]_G$ is the Gerstenhaber bracket (\[Gerst\]), and $m$ is the $A_{\infty}$-structure on $A$ obtained from $Q^{{{\mathfrak{o}}}}$, that is $$\label{m}
m(a_1, a_2, \dots, a_n) = Q^{{{\mathfrak{o}}}} (a_1, a_2, \dots, a_n)\,.$$

Now we define the desired $L_{\infty}$-morphism from $V$ to ${C^{\bullet}}(A,A)$ by the formula: $$\label{define-U}
U(v_1, \dots, v_k)(a_1, \dots, a_n) =
Q^{{{\mathfrak{o}}}} (v_1, \dots, v_k; a_1, \dots a_n)\,,$$ where $v_i \in V$ and $a_j \in A$.

We claim that equation (\[UandQ\]) follows from $$\label{Q-nilpotent}
{\hat{Q}}^{{{\mathfrak{o}}}} \circ {\hat{Q}}^{{{\mathfrak{o}}}}  = 0\,.$$

Indeed, applying the composition $Q^{{{\mathfrak{o}}}} \circ  {\hat{Q}}^{{{\mathfrak{o}}}}$ to the monomial $$(v_1, v_2, \dots, v_k, a_1, a_2, \dots, a_n)$$ with $k\ge 1$ and using (\[Q-hQ\]) we get $$\label{Q-nilpotent-apply}
\begin{array}{c}
\displaystyle
\sum_{p = 1}^k \sum_{{{\lambda}}\in {{\rm Sh}}_{p,k-p}}
(-1)^{{{\varepsilon}}_{{{\lambda}}}} Q^{{{\mathfrak{o}}}} (Q^{{{\mathfrak{c}}}} (v_{{{\lambda}}(1)}, \dots, v_{{{\lambda}}(p)}), v_{{{\lambda}}(p+1)},
\dots, v_{{{\lambda}}(k)} ; a_1, \dots, a_n ) +
\end{array}$$ $$\sum_{1\le t \le s \le n}^{0 \le p \le k}
\sum_{{{\lambda}}\in {{\rm Sh}}_{p,k-p}}
(-1)^{{{\varepsilon}}_{{{\lambda}}} + {{\varepsilon}}^{{\lambda}}_{t,p}}
Q^{{{\mathfrak{o}}}}
(v_{{{\lambda}}(1)}, \dots, v_{{{\lambda}}(p)}, ;
a_1, \dots, Q^{{{\mathfrak{o}}}}(v_{{{\lambda}}(p+1)},
\dots, v_{{{\lambda}}(k)}; a_t, \dots, a_s ), \dots, a_n )+$$ $$\sum_{1\le t \le n+1}^{0 \le p \le k-1}
\sum_{{{\lambda}}\in {{\rm Sh}}_{p,k-p}}
(-1)^{{{\varepsilon}}_{{{\lambda}}} + {{\varepsilon}}^{{{\lambda}}}_{t,p}}
Q^{{{\mathfrak{o}}}}
(v_{{{\lambda}}(1)}, \dots, v_{{{\lambda}}(p)}, ;
a_1, \dots, a_{t-1}, Q^{{{\mathfrak{o}}}}(v_{{{\lambda}}(p+1)},
\dots, v_{{{\lambda}}(k)}), a_{t}, \dots, a_n )\,,$$ where ${{\varepsilon}}_{{{\lambda}}}$ and ${{\varepsilon}}^{{{\lambda}}}_{t,p}$ are defined 