Functors arising from a relational Grothendieck construction

[Luc Pellissier <luc.pellissier@lipn.univ-paris13.fr>]

Dear Category Theorists,

with my adviser Damiano Mazza and his other student Pierre Vial, we are looking
for a name – or even better, a reference – for the following kind of functors:

Let C and B be two categories, F : C ---> D a functor satisfying, for all
morphisms f:c -> c' in C:
- if Ff = g \circ h, then there exists two morphisms k,l such that
  + f = k \circ l 
  + Fk = g
  + Fl = h
- if Ff = id_a for a certain object a, then f itself is an identity.

These functors arise when applying the Grothendieck construction to relational
presheaves: P : B ---> Rel. Indeed, the category of relational presheaves on B
is equivalent (through the Grothendieck construction) to a category whose
objects are such functors over B.

If anyone could point us in a right direction, it would be much appreciated.

Best,

— Luc

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