Connected Functors Peter Johnstone has asked during the last PSSL for a characterization of functors p: E -> B which are "connected" in the sense that the functor Set^B -> Set^E of composition with p is fully faithful. We have found two necesary and sufficient conditions; in the following E and B are arbitrary small categories. Theorem 1. A functor p: E -> B is connected iff every object X of B is an absolute limit of the diagram of all arrows X -> p(Z) for Z ranging through E . Theorem 2. A functor p: E -> B is connected iff for every morphism x: X -> X' of B the category of all factorizations of x through objects of p[E] is connected. More precisely, in Thm 1 we form the diagram of all arrows X -> p(Z) and all E-morphisms whose p-image forms a commutative triangle in B. Then X is equipped with a canonical cone of that diagram; this cone is requested to be an absolute limit. In Thm 2 we consider the category of all triples (Z,q,m) where Z is an object of E and m,q are morphisms of B with x = q.m (and morphisms between these triples are the E-morphisms whose p-images form two commutative triangles in B ). Connectedness of that category has been, for the case of x = id , observed as a necessary condition by Peter. 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 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx