Concerning Ronnie wanderings about terminology around the word FINAL, the following is pertinent: I am just writing a paper with Luis Espannol where we need to develop (the basic part of the theory of cartesian and cocartesian arrows) for families we use the following terminology: consider a functor U: C ---> S, then: 1) a family in C Z _i ---> X over R_i ---> S is FINAL iff: given S ---> T = UY such that there exists Z_i --->Y over R_i ---> S ---> T (that is, R_i ---> S ---> T lifts), then there exists a unique X ---> Y over S ---> T (that is, S ---> T lifts). For topological spaces this is the usual Bourbaki notion of final topology. When U is not understood, we call this "U-FINAL" Notice that for single arrows, we have (proved in the SGA on fibered categories) Z ---> X is final iff it is cocartesian and cocartesian arrows compose 2) a family in C Z _i ---> X over R_i ---> S is SURJECTIVE iff: the family R_i ---> S is an strict (or regular) epimorphic family in S Our aim is to prove under some natural and minimal assumptions: Z _i ---> X is strict epimorphic iff it is final surjective All this is already done Here the leading examples are the topological spaces and the quasitopological spaces in the sense of Spanier (and the whole theory of concrete quasitopoi over S = Sets)