Let $C$ be a symmetric monoidal category and $\mathcal{P}$ an operad in $C$. A morphism of operads $\mathcal{P} \to \mathcal{Q}$ gives rises to a $(\mathcal{P},\mathcal{P})$-bimodule structure on $\mathcal{Q}$ seen as a symmetric collection. We thus get a forgetful functor : $$ \mathcal{P} \downarrow C\mathcal{Op} \to \mathcal{P}Mod\mathcal{P}$$ Where $\mathcal{P} \downarrow C\mathcal{Op}$ is the category of operads under $\mathcal{P}$ (operads $\mathcal{Q}$ with a morphism $\mathcal{P} \to \mathcal{Q}$) and $\mathcal{P}Mod\mathcal{P}$ is the category of $(P,P)$-bimodules.
Are there conditions under which there exists a left adjoint to this forgetful functor ? For example when $\mathcal{P}$ is trivial this simply gives the free-forgetful adjunction between operads and symmetric sequences.
I am reading "Modules over Operads and Functors" by B.Fresse but haven't been able to find a discussion about such a potential adjunction.