Koszul duality and the Poincar\'e-Birkhoff-Witt theorem

Using a homotopy introduced by de Wilde and Lecomte and homological perturbation theory for A_∞-algebras, we give an explicit proof that the universal enveloping algebra $UL$ of a differential graded Lie algebra L is Koszul, via an explicit contracting homotopy from the cobar construction Ω CL of the Chevalley-Eilenberg chain coalgebra $CL$ of L to $UL$.