algebra of ${{\bf s}}^{-2} V$).

The main hero of this article is the DG operad ${{\rm Cobar}}({{\bf S}})$. Since ${{\rm Cobar}}({{\bf S}})$ extends the operad governing homotopy Gerstenhaber algebras we denote it by ${{\bf Ger}}^{+}_{\infty}$.

Let us recall that, due to Proposition 2.15 in [@GJ], a ${{\bf Ger}}^{+}_{\infty}$-algebra structure on $(V,A)$ is a degree $1$ codifferential ${\hat{Q}}$ of the cofree coalgebra ${{\bf S}}(V,A)$.

This observation motivates the definition of a ${{\bf Ger}}_{\infty}^+$-morphism.

\[dfn:Ger-plus-morph\] Let $(V,A)$ and $(V',A')$ be two ${{\bf Ger}}_{\infty}^+$-algebras. A [*${{\bf Ger}}_{\infty}^+$-morphism*]{} from $(V,A)$ to $(V',A')$ is a morphism of ${{\bf S}}$-coalgebras $${\hat{T}}: {{\bf S}}(V,A) \to {{\bf S}}(V',A')$$ with the codifferentials ${\hat{Q}}$ and ${\hat{Q}}'$ corresponding to the ${{\bf Ger}}^+_{\infty}$-structures on $(V,A)$ and $(V', A')$, respectively.

Since ${{\bf S}}(V',A')$ is cofree every morphism $${\hat{T}}: {{\bf S}}(V,A) \to {{\bf S}}(V',A')$$ of ${{\bf S}}$-coalgebras is uniquely determined by its composition $$T = p \circ {\hat{T}}: {{\bf S}}(V,A) \to V' \oplus A'$$ with the projection $p:  {{\bf S}}(V',A') \to  V' \oplus A'$.

It is not hard to see that the compatibility of ${\hat{T}}$ with the codifferentials ${\hat{Q}}$ and ${\hat{Q}}'$ implies that the restrictions: $$\label{T-1-V}
T \Big|_{V} : V \to V'$$ and $$\label{T-1-A}
T \Big|_{A} : A \to A'$$ are morphisms of cochain 