  In the dictionary between the language of (algebraic integrable) connections
and that of (algebraic) $\cD$-modules, to compare the definitions of inverse
images for connections and $\cD$-modules is easy. But the comparison between
direct images for connections (the classical construction of the Gauss-Manin
connection for smooth morphisms) and for $\cD$-modules, although known to
specialists, has been explicitly proved only recently in a paper of Dimca,
Maaref, Sabbah and Saito in 2000, where the authors' main technical tool was M.
Saito's equivalence between the derived category of $\cD$-modules and a
localized category of differential complexes. The aim of this short paper is to
give a simplified summary of the [DMSS] argument, and to propose an alternative
proof of this comparison which is simpler, in the sense that it does not use
Saito equivalence. Moreover, our alternative strategy of comparison works in a
context which is a precursor to the Gauss-Manin connection (at the level of
$f^{-1}\cD_Y$-modules, for a morphism $f:X\to Y$), and may be of some intrinsic
interest.
