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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]