A\otimes A  \ar[r]^-{1\otimes {{\Delta}}_{{{\mathfrak{o}}}}} \ar[d]_{\rho \otimes 1}  &
A \otimes A \otimes A \ar[r]^{\rho \otimes \rho \otimes 1}
\ar[d]_{\rho\otimes 1 \otimes 1} &
V \otimes V \otimes A \\
V \otimes A \ar[r]^-{1 \otimes {{\Delta}}_{{{\mathfrak{o}}}}} &
V \otimes A  \otimes A \ar[ur]_{1 \otimes \rho \otimes 1} &  ~
}$$

The corresponding statement for $\mu_r$ can be proved in a similar way.

Let us denote by ${{\bf S}}$ the cooperad which governs the ${{\bf S}}$-coalgebras. This is a 2-colored cooperad. We reserve the color ${{\mathfrak{c}}}$ for vectors in $V$ and the color[^2] ${{\mathfrak{o}}}$ for vectors in $A$. In Subsection \[subsect:S-versus-sc\] we describe a link between the cooperad ${{\bf S}}$ and the cohomology cooperad $H^{{{\bullet}}}({{\rm SC}})$ of Voronov’s Swiss Cheese operad ${{\rm SC}}$.

The cofree ${{\bf S}}$-coalgebra ${{\bf S}}(V,A)$ cogenerated by the pair $(V,A)$ is the direct sum $$\label{bS-V-A}
{{\bf S}}(V, A) = {{\bf S}}^{{{\mathfrak{c}}}}(V, A) \,\oplus\,
{{\bf S}}^{{{\mathfrak{o}}}}(V, A)$$ where $$\label{bS-V-A-mc}
{{\bf S}}^{{{\mathfrak{c}}}}(V, A) = {{\bf Ger}}^{\vee}(V)$$ and $$\label{bS-V-A-ma}
{{\bf S}}^{{{\mathfrak{o}}}}(V, A) =
{{\bf s}}\,  \big( {{\underline{S}}}({{\bf s}}^{-2} V) \otimes   T({{\bf s}}^{-1}A)  \big)
\, \oplus  \, {{\bf s}}\, {{\underline{T}}}({{\bf s}}^{-1} A)\,,$$ where ${{\underline{T}}}({{\bf s}}^{-1}A)$ (resp. ${{\underline{S}}}({{\bf s}}^{-2} V)$) is the truncated tensor algebra of ${{\bf s}}^{-1} A$ (resp. the truncated symmetric 