i wrote:

[this “restoration of exactness”, the fact that cauchyness spreads from merely the finite discrete colimits to much more general (homotopy-)colimits when you pass from abelian groups to (“2-sided”) chain complexes, means that the (weak) cauchy completion of a dg algebroid is almost the same thing as either its free completion or its free cocompletion.]

but even though i think that i remember a lot of what i was trying to say here, i'm confused at the moment about why the free completion and the free cocompletion (which indeed should be equivalent to each other) should also be equivalent to the "(weak) cauchy completion" ... ??oh, wait, no, maybe it's obvious ... that's the whole point ... that the definition of "cauchy-completion" is that it should include just those limits and colimits which can be indifferently thought of as either one ... ??? ....