s}}^{-1} {{\bf S}}^{{{\mathfrak{o}}}}(V, A) = {{\underline{S}}}({{\bf s}}^{-2} V) \otimes  T({{\bf s}}^{-1}A) ~ \oplus ~
 {{\underline{T}}}({{\bf s}}^{-1} A )$$ and conjugate the ${{\mathfrak{o}}}$-component ${\hat{Q}}^{{{\mathfrak{o}}}}$ of the codifferential ${\hat{Q}}$ by the suspension isomorphism. (Thanks to this technical modification we will have simpler expressions for sings in our formulas.) We will use the same notation $Q^{{{\mathfrak{o}}}}$ for this conjugated map: $$\label{Q-ma-conjugated}
Q^{{{\mathfrak{o}}}} :   \, \big(\, {{\underline{S}}}({{\bf s}}^{-2} V) \otimes   T({{\bf s}}^{-1}A) \, \big)
\, \oplus  \, {{\underline{T}}}({{\bf s}}^{-1} A)
\to {{\bf s}}^{-1} \, A\,.$$

Due to the above shift the comultiplication ${{\Delta}}_{{{\mathfrak{o}}}}$ on (\[desusp-bS-ma\]) has degree $0$ and the explicit formula for ${{\Delta}}_{{{\mathfrak{o}}}}$ reads $$\label{D-ma}
\begin{array}{c}
\displaystyle
{{\Delta}}_{{{\mathfrak{o}}}} (v_1, \dots, v_k; a_1, \dots, a_n) = \\[0.3cm]
\displaystyle
\sum_{(p,t) \in E_{k,n}}
\sum_{{{\lambda}}\in {{\rm Sh}}_{p,k-p}} (-1)^{\eta^{{{\lambda}}}_{p,t}}
(v_{{{\lambda}}(1)}, \dots, v_{{{\lambda}}(p)}; a_1, \dots, a_t)
\otimes \\[0.3cm]
\displaystyle
(v_{{{\lambda}}(p+1)}, \dots, v_{{{\lambda}}(k)}; a_{t+1}, \dots, a_n)
\end{array}$$ where $$E_{k,n} = \{0,1, \dots, k\} \times  \{0,1, \dots, n\}
\setminus \{(0,0), (k,n)\}\,,$$ $$\label{eta-la-p-t}
\eta^{{{\lambda}}}_{p,t} =  \sum_{i<j ~ {{\lambda}}(i) > {{\lambda}}(j)}
|v_i||v_j |  ~ +  ~ \sum_{i=1}^t \sum_{j=p+1}^k
|v_{{{\lambda}}(j)}| (|a_i| + 1)\,.$$

Using the compatibility of ${\hat{Q}}$ with the ${{\bf S}}$-coalgebra structure one deduces readily a formula for ${\hat{Q}}^{{{\mathfrak{o}}}}$ in terms of $Q^{{{\mathfrak{c}}}}$ and $Q^{{{\mathfrak{o}}}}$: $$\label{Q-hQ}
\begin{array}{c}
\displaystyle
{\hat{Q}}^{{{\mathfrak{o}}}}(v_1, \dots, v_k; a_1, \dots, 