Is there an already studied notion of $\infty$-morphism between algebras over a quasi-free operad $P = (T(E), \partial)$?
If the operad $P$ is Koszul, or of the form $\Omega C$ for $C$ a cooperad, this is already well-defined and studied (see e.g. Loday, Vallette - Algebraic Operads). However, what about the more general context in which we only require $P$ to be quasi-free? I would expect it to be defined analogously: since $P$ is quasi-free, $C := s E$ is endowed with a homotopy cooperad structure, and hence one can define an $\infty$-morphism as a codifferential between cofree homotopy(?) coalgebras $C(A) \to C(B)$. Does this make sense? Is it already done?
