Discrete fibrations vs. functors into Set

[Uwe Egbert Wolter <Uwe.Wolter@uib.no>]

Dear all,

We consider two categories. The first category with objects given by a
small category B and a functor F:B->Set and morphisms
(H,alpha):(B,F)->(C,G) given by a functor H:B->C and a natural
transformation alpha:F=>H;G. The second category has as objects discrete
fibrations p:E->B and morphisms (H,phi):(E,p)->(D,q:D->C) are given by
functors H:B->C and phi:E->D such that phi;q=p;H.

1. Are there any "standard" terms and notations for these categories?
2. For both categories we do have projection functors into Cat! Are
these functors kind of (op)fibrations?
3. We know that the Grothendieck construction establishes equivalences
between corresponding fibers of the two projection functors into Cat. Do
these fiber-wise equivalences extend to an equivalence between the two
categories?

Thanks

Uwe



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