Note that the suggestion below is standard terminology since about 30 years. See also Adamek, Herrlich, Strecker: Abstract and concrete categories; Wiley 1990 (also available at http://katmat.math.uni-bremen.de) H.-E. Porst Am 26.12.2005 um 22:57 schrieb Eduardo Dubuc:
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"
-- Hans-E. Porst porst@uni-bremen.de Bremen, Germany Fax: +49-421-75643