Let $\mathrm{Ass} $ denote the operad, whose algebras are associative unital algebras, considered as a dg-operad. Denote $\mathrm{Ch} $ the category of chain complexes over a commutative ring $\mathrm{R} $ and denote $\mathrm{Bialg}_{ } (\mathrm{Ch} )$ the category of cocommutative (counital and unital) bialgebras in $\mathrm{Ch}. $

Let $\mathcal{O}$ be a dg-operad and $\phi: \mathcal{O} \to \mathrm{Ass} $ a map of dg-operads. $\phi$ induces a free-forgetful adjunction $ \mathcal{U} : \mathrm{Alg}_{ \mathcal{O} } ( \mathrm{Ch} ) \rightleftarrows \mathrm{Alg}_{ \mathrm{Ass} } ( \mathrm{Ch} ).$

When does the left adjoint $ \mathcal{U} : \mathrm{Alg}_{ \mathcal{O} } ( \mathrm{Ch}) \to \mathrm{Alg}_{ \mathrm{Ass} } (\mathrm{Ch} ) $ lift to a functor $ \mathrm{Alg}_{ \mathcal{O} } ( \mathrm{Ch} ) \to \mathrm{Bialg}_{ } (\mathrm{Ch} )? $

For $\mathcal{O}$ the Lie operad there is such a lift.

For $\mathcal{O}$ the Lie operad the left adjoint $\mathcal{U}$ is the enveloping algebra. For every Lie-algebra $\mathrm{X}$ the bialgebra structure on $\mathcal{U}(\mathrm{X})$ is encoded in a symmetric monoidal structure on the category $\mathrm{LMod}_{ \mathcal{U}(\mathrm{X}) } (\mathrm{Ch} )$ (being the category of representations of $\mathrm{X}$) lifting the symmetric monoidal structure on $\mathrm{Ch} .$

The category of representations of $\mathrm{X}$ can also be described as the category $\mathrm{Mod}^{\mathcal{O}}_{ \mathrm{X} } (\mathrm{Ch} ) $ of $\mathrm{X}$-modules over the Lie-operad.

This leads to my 2. question: For which dg-operads $ \mathcal{O}$ and $ \mathcal{O}$-algebras $\mathrm{X}$ does the symmetric monoidal structure on $\mathrm{Ch} $ lift to a symmetric monoidal structure on $\mathrm{Mod}^{\mathcal{O}}_{ \mathrm{X} } (\mathrm{Ch} ) ?$

More generally given a dg-operad $\mathcal{O}$ and a $\mathcal{O}$-algebra $\mathrm{X}$ there is a enveloping associative algebra $\mathcal{U}_{ \mathcal{O}}( \mathrm{X} ) $ with the property that we have a canonical equivalence $$\mathrm{LMod}_{\mathcal{U}_{ \mathcal{O}}( \mathrm{X} ) } (\mathrm{Ch} ) \simeq \mathrm{Mod}^{\mathcal{O}}_{ \mathrm{X} } (\mathrm{Ch} ) .$$

As a cocommutative bialgebra structure on an associative algebra corresponds to a symmetric monoidal structure on its category of left modules lifting the symmetric monoidal structure on $ \mathrm{Ch} $, my 2. question is equivalent to the following:

For which dg-operads $ \mathcal{O}$ does the enveloping associative algebra functor $ \mathcal{U} : \mathrm{Alg}_{ \mathcal{O} } ( \mathrm{Ch}) \to \mathrm{Alg}_{ \mathrm{Ass} } (\mathrm{Ch} ) $ lift to a functor $ \mathrm{Alg}_{ \mathcal{O} } ( \mathrm{Ch} ) \to \mathrm{Bialg}_{ } (\mathrm{Ch} )? $