General idea: \(R{\hbox{-}}\)modules \(M\) can be specified by \(S\otimes_R M\) along with descent data.

Refs: stacks vector bundle descent data

faithfully flat descent : there is an equivalence of categories \({\mathsf{R}{\hbox{-}}\mathsf{Mod}} \to {\mathsf{Desc}}(R\searrow S)\),

The problem is that I don’t really know how to relate the bottom line (whose exactness is the usual condition for sheaves, stacks, etc) to the intermediate steps. This seems like it wants \({\mathcal{F}}({\textstyle\coprod}{-}) = \prod {\mathcal{F}}({-})\), so it commutes with (co?)limits, since probably contravariant functors send coproducts to products. Moreover the bar construction in the 2nd line might form a simplicial object? And the condition of satisfying descent is maybe related to either this being a simplicial object, or its image in the bottom line assembling to a simplicial object, since there are clear degeneracy maps and one would want sections in order to build face maps. Super vague, there are a lot of details missing here!!