  For a morphism of smooth schemes over a regular affine base we define
functors of derived direct image and extraordinary inverse image on coderived
categories of DG-modules over de Rham DG-algebras. Positselski proved that for
a smooth algebraic variety $X$ over a field $k$ of characteristic zero the
coderived category of DG-modules over $\Omega^\bullet_{X/k}$ is equivalent to
the unbounded derived category of quasi-coherent right ${\mathscr
D}_X$-modules. We prove that our functors correspond to the functors of the
same name for ${\mathscr D}_X$-modules under Positselski equivalence.
