  We study deformations of Fourier-Mukai transforms in general complex analytic
settings. We start with two complex manifolds X and Y together with a coherent
Fourier-Mukai kernel P on their product. Suppose that P implements an
equivalence between the coherent derived categories of X and Y. Given an
arbitrary formal quantization of X we construct a unique quantization of Y such
that the Fourier-Mukai transform deforms to an equivalence of the derived
categories of the quantizations. Here quantizations are understood in the
framework of stacks of algebroids.
