Koszul duality and the Poincaré-Birkhoff-Witt theorem

Using a homotopy introduced by de Wilde and Lecomte and homological perturbation theory for $A_{\mathrm{\infty }}$-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 $\mathrm{\Omega }CL$ of the Chevalley-Eilenberg chain coalgebra $CL$ of $L$ to $UL$.