Extension of functors for algebras of formal deformation

Suppose we are given complex manifolds X and Y together with substacks S and S′ of modules over algebras of formal deformation A on X and A′ on Y, respectively. Also, suppose we are given a functor Φ from the category of open subsets of X to the category of open subsets of Y together with a functor F of prestacks from S to S′ ○ Φ. Then we give conditions for the existence of a canonical functor, extension of F to the category of coherent A-modules such that the cohomology associated to the action of the formal parameter h{stroke} takes values in S. We give an explicit construction and prove that when the initial functor F is exact on each open subset, so is its extension. Our construction permits to extend the functors of inverse image, Fourier transform, specialisation and micro-localisation, nearby and vanishing cycles in the framework of D{script}[[h{stroke}]]-modules. We also obtain the Cauchy-Kowalewskaia-Kashiwara theorem in the non-characteristic case as well as comparison theorems for regular holonomic D{script}[[h{stroke}]]-modules and a coherency criterion for proper direct images of good D{script}[[h{stroke}]]- modules.