Recently it has been discussed what is the appropriate notion of connecteness for a category \X relative to a category \B. The following appears as natural to me. Let \B be a category and P : \X -> \B be a fibration of categories with a terminal object and with internal sums. Then for every object I in \B there is an obvious functor \Delta_I : \B/I -> \X_I sending u : J -> I to \coprod_u 1_J. An object X \in \X_I is an I_indexed family of connected objects iff the functor \X_I(X,\Delta_I(-)) : (\B/I)^\op -> Set is represented by \id_I, i.e there exists eta_X : X \to 1_I such that for every cocartesian arrow \phi : 1_J -> \Delta_I(u) over u : J -> I and vertical arrow \alpha : X -> \Delta_I(u) there exists a unique arrow s : I -> J making the diagram X --------------------> 1_I | | | \alpha | 1_s | | V cocart. V \Delta_I(u) <---------------1_J commute (where the top arrow is vertical). In case I = 1 (where we write \Delta for \Delta_I) this means that for every f : X -> Delta(I) there is a unique i : 1 -> I with f = \Delta(i) \circ eta_X. Notice that in case \B has finite limits, P is a fibration of categories with finite limits and stable and disjoint sums the fibration P is equivalent to \Delta^* P_{\X_1}. This is an old result of Moens (1982) (see section 15 of www.mathematik.tu-darmstadt.de/~streicher/FIBR/FibLec.pdf.gz for an exposition). Thomas