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This paper is devoted to a new proof of the comparison between the derived direct image functor for $\cD$-Modules and the construction of the Gauss-Manin connection for smooth morphisms. Moreover, our alternative strategy of comparison works in a context whichis 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

Algebraic connections vs. algebraic $\mathscr D$-modules: inverse and direct images