Since ${\hat{U}}$ is a morphism to a cofree coalgebra, it is uniquely determined by its composition $$U = p \circ {\hat{U}}:   {{\bf s}}^2 S ({{\bf s}}^{-2} {{\cal L}}) \to {{\widetilde{\mathcal L}}}$$ with the natural projection $$p:   {{\bf s}}^2 S ({{\bf s}}^{-2} {{\widetilde{\mathcal L}}}) \to {{\widetilde{\mathcal L}}}\,.$$

The notation ${{\bf Ger}}$ is reserved for the operad governing Gerstenhaber algebras [@Ger], [@GJ] and ${{\bf Ger}}^{\vee}$ denotes the corresponding Koszul dual [@Fresse], [@GJ], [@GK] cooperad. This cooperad can be obtained from the linear dual ${{\bf Ger}}^*$ of ${{\bf Ger}}$ by applying the operation ${{\Lambda}}$ twice: $${{\bf Ger}}^{\vee} = {{\Lambda}}^2 {{\bf Ger}}^*\,.$$ Following [@Fresse], [@GJ], and [@GK] the DG operad ${{\rm Cobar}}({{\bf Ger}}^{\vee})$ can be chosen as a model for the operad which governs homotopy Gerstenhaber algebras. So, in this paper, we set $${{\bf Ger}}_{\infty} = {{\rm Cobar}}({{\bf Ger}}^{\vee})\,.$$

For colored (co)operads we use notation and conventions from paper [@BM-colors] by C. Berger and I. Moerdijk. All colored (co)operads in this paper are $2$-colored and we denote the two colors by ${{\mathfrak{c}}}$ and ${{\mathfrak{o}}}$. Furthermore, we fix the order ${{\mathfrak{c}}}< {{\mathfrak{o}}}$ on the set $\{{{\mathfrak{c}}}, {{\mathfrak{o}}}\}$. Hence, due to [@BM-colors Remark 1.3], every $2$-colored (co)operad ${{\cal O}}$ is completely determined by objects $${{\cal O}}(\underbrace{{{\mathfrak{c}}}, 