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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]