CAUTION: The Sender of this email is not from within Dalhousie. 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]