CAUTION: The Sender of this email is not from within Dalhousie. Answer to Uwe, The notion of a discrete fibration, and its equivalence with a functor to Sets as well as with the action of a category on a set,?? were initially introduced by Charles Ehresmann in "Gattungen von Lokalen Strukturen" (Jahresbericht d. DMV Bd. 60 (1957) S. 4 9 ??? 7 7, reprinted in https://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/Ehresmann_C.-Oeuvres_I... <https://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/Ehresmann_C.-Oeuvres_II_1.pdf> The first category you describe is generally called the category of diagrams into Sets, Diag(Sets); the second one which is isomorphic, is called the category of (morphisms between) discrete fibrations. The category Diag(Sets), and more generally the (2-)category Diag(H) for any category H, have been extensively studied by Ren?? Guitart in some 1970's papers, in particular in D??compositions et lax-compl??tions, (avec L. Van den Bril), CTGD XVIII,4, p. 333-407, 1977. http://archive.numdam.org/article/CTGDC_1977__18_4_333_0.pdf <http://archive.numdam.org/article/CTGDC_1977__18_4_333_0.pdf> In the last years, with Alexandre Popoff, C. Agon and M. Andreatta, we have studied and applied Diag(H) in papers on Math/Music theory, naming its objects "Poly-Klumpenhouwer-Nets" (or PK-Net) with values in H, for instance in "From Nets to PK-Nets: a categorical approach", /Perspective of new music /54-2, 2016, 5-68. For other. references, consult my personal site https://ehres.pagesperso-orange.fr/ <https://ehres.pagesperso-orange.fr/> Kind regards Andr??e [For admin and other information see: http://www.mta.ca/~cat-dist/ ]