a variant of the one in (Nielsen 2004, TAC 12(7), pp 248???261), but consider

You mean Niefield and not Nielsen. This paper makes clear the relation to Giraud-Conduch'e functors. But they just study the Weak Factorization Lifting Property (WFLP) which in terms of distributors means that all components of the natural transformation corresponding to lax preservation of composition are surjective. Faithful functors to B reflecting identities correspond to "relational variable sets" on B as described in the Niefield paper. But they are NOT Conduch'e fibrations since they just validate WFLP and not FLP. p : E -> B is a Conduch'e fibration (i.e. validatates FLP) iff it is exponentiable in Cat/B but p validates WLFP iff it is exponentiable in Cat_f/B (Cor.4.2 in Niefield paper). What Niefield calls Grothendieck construction is an instance of the transition from a lax normalised functor from B^op to Dist to a functor to B (due to Benabou). But this has nothing to do with what you describe as Grothendieck construction which rather is chanke of base along a functor B* -> B. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]