Dear Luc, by a theorem of B'enabou (and R. Street independently I guess) functors to BB correspond to lax normalized functors from BB^op to Dist, the bicategory of distributors. (see e.g. "Distributors at Work" on my homepage). This equivalence restricts to one between Conduch'e fibrations over BB and normalized pseudofunctors from BB^op to Dist. Replacing Set by 2 (i.e. {\emptyset,{\emptyset}}) and restricting to discrete guys functors from BB^op to Rel are equivalent to Conduch'e fibrations over BB which are faithful and reflect identities. In this case the Conduch'e condition amounts to a unique lifting property of factorization from the base tot he total category. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]