All known realizations of complete Lie algebras coincide

We prove that for any reduced differential graded Lie algebra L, the classical Quillen geometrical realization $⟨L{⟩}_Q$ is homotopy equivalent to the realization $⟨L⟩=Ho{m}_{\mathbf{c}\mathbf{d}\mathbf{g}\mathbf{l}}({\mathfrak{L}}_{\bullet },L)$ constructed via the cosimplicial free complete differential graded Lie algebra ${\mathfrak{L}}_{\bullet }$. As the latter is a deformation retract of the Deligne-Getzler-Hinich realization MC${}_{\bullet }(L)$ we deduce that, up to homotopy, there is only one realization functor for complete differential graded Lie algebras. Immediate consequences include an elementary proof of the Baues-Lemaire conjecture and the description of the Quillen realization as a representable functor.