  We construct quantization functors of quasi-Lie bialgebras. We establish a
bijection between this set of quantization functors, modulo equivalence and
twist equivalence, and the set of quantization functors of Lie bialgebras,
modulo equivalence. This is based on the acyclicity of the kernel of the
natural morphism from the universal deformation complex of quasi-Lie bialgebras
to that of Lie bialgebras. The proof of this acyclicity consists in several
steps, ending up in the acyclicity of a complex related to free Lie algebras,
namely, the universal version of the Lie algebra cohomology complex of a Lie
algebra in its enveloping algebra, viewed as the left regular module. Using the
same arguments, we also prove the compatibility of quantization functors of
quasi-Lie bialgebras with twists, which allows us to recover our earlier
results on compatibility of quantization functors with twists in the case of
Lie bialgebras.
