Not a good title.
Suppose we have two dg symmetric Koszul operads, say $O_1$ and $O_2$. Then their (homotopy) algebras over a dg vector space $V$ are (equivalent to) twisting morphisms
$$f\in TW(O^*_{i}, End_V)$$
where $O^*$ is the Koszul dual dg-cooperad. (I don't even know the \TeX symbol for the anti-shriek) and $TW(O^*_{i}, End_V)$ is the solution set of the Maurer Cartan equation with respect to the dg Lie algebra structure on $Hom_{S-mod}(O^*_{i}, End_V)$.
Now any (infinity) morphism of dg Lie algebras
$$G: Hom_{S-mod}(O^*_1, End_V) \to Hom_{S-mod}(O^*_2, End_V)$$
maps solutions of the Maurer Cartan equation into solutions of the Maurer Cartan equation, therefore transforming (homotopy) $O_1$-algebras into (homotopy) $O_2$-algebras.
One way to find some of these dg Lie algebra maps, is via cooperad morphisms: Suppose we have an (infinity) morphism of dg-cooperads
$$F: O^*_2 \to O^*_1.$$
Then $F$ induces an (infinity) morphism of dg-Lie algebras
$$G_F:= F^*: Hom_{S-mod}(O^*_1, End_V) \to Hom_{S-mod}(O^*_2, End_V)$$
via pull-back.
Now my question is:
Is any (infinity) Lie algebra morphism $G$ induced by some (infinity) cooperad morphism $F$? I mean is there a 1:1 correspondence? Or are there (infinity) dg-Lie algebra maps $G$, that do not come from some (infinity) dg-cooperad map $F$?