  We give simple explicit formulas for deformation quantization of Poisson-Lie
groups and of similar Poisson manifolds which can be represented as moduli
spaces of flat connections on surfaces. The star products depend on a choice of
Drinfe\v{l}d associator and are obtained by applying certain monoidal functors
(fusion and reduction) to commutative algebras in Drinfe\v{l}d categories. From
a geometric point of view this construction can be understood as a quantization
of the quasi-Poisson structures on moduli spaces of flat connections.
