The connected functors we have mentioned the other day, i.e. those functors p: E -> B for which (-).p Set^B -> Set^E is fully faithful, are easily seen to be precisely the lax epimorphisms of the 2-category Cat (of small categories). We do not know whether every pseudoepimorphism in Cat is connected, does anyone know? The answer is affirmative if B is a preorder. J.Adamek, R. El Bashir, M. Sobral and J. Velebil xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx