Compatibility of quantization functors of Lie bialgebras with duality and doubling operations

We study the behavior of the Etingof-Kazhdan quantization functors under the natural duality operations of Lie bialgebras and Hopf algebras. In particular, we prove that these functors are "compatible with duality", i.e., they commute with the operation of duality followed by replacing the coproduct by its opposite. We then show that any quantization functor with this property also commutes with the operation of taking doubles. As an application, we show that the Etingof-Kazhdan quantizations of some affine Lie superalgebras coincide with their Drinfeld-Jimbo-type quantizations.