# Riemann Hilbert Correspondence with fixed stractification

Riemann Hilbert Correspondence states for complex manifold $$X$$, the bounded derived category of $$D$$ modules on $$X$$ with cohomology being regular holonomic is equivalent to the bounded derived category of sheaves on $$X$$ with constructible cohomology.

My question is if we fix a particular stractification $$\mathcal{S}$$ for $$X$$ and require the cohomology sheaves is constructible with $$\mathcal{S}$$, what would be the corresponding subcategory on the $$D$$ modules side?

First, to be a local system for a constructible sheaf translates under RH to the D-module being smooth (i.e., free of finite rank as an $$\mathcal{O}$$-module).
So, if constructibility of a sheaf $$F$$ means that for any stratum $$i_{\alpha} : S_{\alpha} \to X$$ one has that $$i_{\alpha}^* F$$ is a local system, the corresponding condition for a $$D$$-module $$M$$ will be that $$i_{\alpha}^* M$$ is smooth for all $$\alpha$$.
(Slightly orthogonally to the question itself; One also needs to be careful that there is a second notion - a sheaf $$F$$ is $$!$$-constructible if $$i_{\alpha}^! F$$ is a local system for all $$\alpha$$. In some situations (nice equivariant ones usually), $$!$$-constructibility is equivalent to $$*$$-constructibility.)