In fact there's a more general construction, recorded in the Stacks Project https://stacks.math.columbia.edu/tag/08NF and in the case that the codomain (only!) is fibred in groupoids it shows that any cartesian functor factors as an equivalence of fibrations followed by a fibration. Regards, David On 21 Sep. 2017 7:17 pm, "Thomas Streicher" < streicher@mathematik.tu-darmstadt.de> wrote:
So I guess this extends your example where the codomain is a discrete fibration, merely having to replace the domain by an equivalent category. This makes the original cartesian functor a Street fibration, I believe.
Indeed every functor F between groupoids is a Street fibration and thus equivalent to a Grothendieck fibration in the sense that there is a fibarion P and an equivalence E such that F = PE.
Interesting that this extends to fibered functors!
Thomas
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