  Suppose we are given complex manifolds $X$ and $Y$ together with substacks
$\mathcal{S}$ and $\mathcal{S}'$ of modules over algebras of formal deformation
$\mathcal{A}$ on $X$ and $\mathcal{A}'$ on $Y$, respectively. Suppose also we
are given a functor $\Phi$ from the category of open subsets of $X$ to the
category of open subsets of $Y$ together with a functor $F$ of prestacks from
$\mathcal{S}$ to $\mathcal{S}'\circ\Phi$. Then we give conditions for the
existence of a canonical functor, extension of $F$ to the category of coherent
${\sha}$-modules such that the cohomology associated to the action of the
formal parameter $\hbar$ takes values in $\mathcal{S}$. 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, specialization and microlocalization, nearby
and vanishing cycles in the framework of $\shd[[\hbar]]$-modules. We also
obtain a Cauchy-Kowalewskaia-Kashiwara theorem in the non-characteristic case
as well as comparison theorems for regular holonomic $\shd[[\hbar]]$-modules
and a coherency criterion for proper direct images of good
$\shd[[\hbar]]$-modules.
