Suppose that $\mathcal{P}$ is a connected, unital operad in $\mathbb{k}$-vector spaces (or complexes), i.e. $\mathcal{P}(1)=\mathbb{k}$ and the unit map for $\mathcal{P}$ is the identity. One may form the simplicial bar construction $C^o_*(\mathcal P)$ on $\mathcal{P}$. This produces a simplicial symmetric sequence as follows. For the $n$-simplices, we set $$ C^o_n(\mathcal P) = \mathcal {P}^{\circ n}. $$ The inner face maps are induced from the operad structure on $\mathcal P$ and the outer face maps are induced from the natural augmentation on $\mathcal P$. The degeneracies are induced from the unit map of $\mathcal P$. One may take the chain complex modulo degeneracies of this simplicial symmetric sequence and obtain a dg-symmetric sequence $N(\mathcal P)$.
In this paper, Fresse constructs a map $$ B(\mathcal P) \to N(\mathcal P), $$ where $B(\mathcal P)$ is the operadic bar construction (as in, for example, Loday-Vallette). He shows that it is a quasi-isomorphism of complexes of symmetric sequences. The object $B(\mathcal P)$ is a dg-cooperad. My question is: is $N(\mathcal P)$ also a dg-cooperad, and if so, is the morphism of Fresse a morphism of cooperads? Same question for the dual statements.
My intuition says that this is a cooperad, using the maps $$ \mathcal{P}^{\circ n} \to \bigoplus_{p+q=n} \mathcal{P}^{\circ p} \circ \mathcal{P}^{\circ q}, $$ but I haven't worked through all the checks.
