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Did some reorganizing at Grothendieck fibration.
I don't think "we" have used "prestack" enough to give it a definite meaning. I've always thought that "presheaf of groupoids/categories" would be a much better meaning of "prestack" than "separated presheaf of groupoids/categories" (where by "injective" I presume you mean "fully faithful") but the latter meaning seems to be universal especially in the algebraic geometry community, so I'd be hesitant to depart from it.
Maybe an algebraic geometer can answer your second question better, but I think it's because a lot of the time those are what they see. For instance, if you take a sheaf of sets which is acted on by a sheaf of groups, or more generally an internal groupoid in sheaves, then its "homotopy quotient" is a stack of groupoids. I think most of the stacks that algebraic geometers care about are "moduli stacks" and arise in that way.
Grothendieck construction approach to Baues-Wirsching cohomology can be found in the paper of Petar Pavesic, for whom I created an entry with a link to his (pale scan of the) paper in JPAA. The page is not created with the proper diacritics, which are however properly used inside the entry. It is open to improvements.
Thanks, Zoran, that looks very helpful and goes some way to what I want. I knew Petar back when I used to visit Genoa quite often as he was often there.
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