We know that in Tamarkin's proof of Kontsevich's formality theorem, he defined the $G_\infty$ structure on the Hochschild cochain complex $C^\cdot(A,A)$ and constructed a $G_\infty$ morphism from $HH^\cdot(A,A)$ to $C^\cdot(A,A)$, which makes Kontsevich's original $L_\infty$ morphism as the $L_\infty$ part of it.

So what is the precise definition of "the $L_\infty$ part of a $G_\infty$ morphism"? I know it looks like we just forget the homotopies for the cup product but how to make it precise?