  In this paper, we study the C-enriched pre-Lie operad defined by Calaque and
Willwacher for any Hopf cooperad C to produce conceptual constructions of the
operads acting on various deformation complexes. Maps between Hopf cooperads
lead to maps between the corresponding enriched pre-Lie operads; we prove
criteria for the module action of the domain on the codomain to be free, on the
left and on the right. In particular, this implies a new functorial
Poincar\'e--Birkhoff--Witt type theorem for universal enveloping brace algebras
of pre-Lie algebras.
