Operadic Bar-Cobar Adjunction

Given a dg operad P and a dg cooperad C, we first explain how to equip Hom(C,P) with a dg (pre-)Lie algebra structure.
We then construct the operadic bar and cobar functors between the categories of augmented dg operads and coaugmented
conilpotent dg cooperads and show that they define an adjunction, which is precisely given by the Maurer-Cartan elements
of Hom(C,P).