Lift of $C\mathrm{\_}\mathrm{\infty }$ and $L\mathrm{\_}\mathrm{\infty }$ morphisms to $G\mathrm{\_}\mathrm{\infty }$ morphisms

Let $\text{g}\mathrm{\_}2$ be the Hochschild complex of cochains on $C^{\mathrm{\infty }}({\text{RM}}^n)$ and $\text{g}\mathrm{\_}1$ be the space of multivector fields on ${\text{RM}}^n$. In this paper we prove that given any $G\mathrm{\_}\mathrm{\infty }$-structure ({\rm i.e.} Gerstenhaber algebra up to homotopy structure) on $\text{g}\mathrm{\_}2$, and any $C\mathrm{\_}\mathrm{\infty }$-morphism $\varphi$ ({\rm i.e.} morphism of commutative, associative algebra up to homotopy) between $\text{g}\mathrm{\_}1$ and $\text{g}\mathrm{\_}2$, there exists a $G\mathrm{\_}\mathrm{\infty }$-morphism $\mathrm{\Phi }$ between $\text{g}\mathrm{\_}1$ and $\text{g}\mathrm{\_}2$ that restricts to $\varphi$. We also show that any $L\mathrm{\_}\mathrm{\infty }$-morphism ({\rm i.e.} morphism of Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a $G\mathrm{\_}\mathrm{\infty }$-morphism, using Tamarkin’s method for any $G\mathrm{\_}\mathrm{\infty }$-structure on $\text{g}\mathrm{\_}2$. We also show that any two of such $G\mathrm{\_}\mathrm{\infty }$-morphisms are homotopic.