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Dear Uwe,

I believe that the answer to questions 2. & 3. is yes. It is likewise that both claims follow quite formally from the classical ``local'' correspondence between discrete fibrations/opfibrations and contra/covariant functors on small categories. This correspondence is of course a special case of Grothendieck's correspondence between between fibrations/opfibrations and contra/covariant pseudofunctors.

In the discrete case you are dealing with there is an additional link with Lawvere's comprehension schemes and Street-Walter's comprehensive factorisation of a functor. I thematized this in joint work with Ralph Kaufmann.

The morphisms in your first category just express the fact that the functor P:Cat->CAT which takes a small category A to its diagram category PA=[A,Set] comes equipped, for each f:A->B in Cat, with an adjoint pair f_!:PA<=>PB:f^* given by left Kan extension along f, resp. precomposition with f. Composition in your first category amounts to composition of these adjunctions.

Each of the PA has a terminal object *_A, and one can check that the functor Cat/B->PB which takes f:A->B to f_!(*_A) has a fully faithful right adjoint. This is closely related to Lawvere's "comprehension schemes".

The unit of the latter adjunction replaces a general f:A->B by a discrete opfibration. You get actually the factorisation of f into an initial functor followed by a discrete opfibration (this is dual to Walter-Street's factorisation into final functor followed by discrete fibration), explicitly:

A->el(f_!(*_A))->B

where the category in the middle is the category of elements of the diagram f_!(*_A). The functor f is a discrete opfibration iff the first arrow is invertible, and an inital functor iff the second arrow is invertible. From this you can derive an equivalence between your two categories (answering question 3).

The orthogonal factorisation systems (initial, discrete opfibrations) and (final, discrete fibration) have nice stability properties, nowadays running under the denomination ``modality''.

All the best, Clemens.

Le 2020-12-02 14:53, Uwe Egbert Wolter a ??crit??:

...

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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