  We prove that for any reduced differential graded Lie algebra L, the
classical Quillen geometrical realization $\langle L\rangle_Q$ is homotopy
equivalent to the realization $\langle L\rangle= Hom_{\bf
cdgl}(\mathfrak{L}_\bullet, L)$ 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.
