I am confronted with problems of "contradictory terminology" which I would like to solve and, since english is not my language, I need some suggestions. Let F: Y-----> X be a functor such that for every object x of X the comma category (x,F) is connected.Such functors, although they are not defined in all generality, are called "cofinal" in SGA 4 , and "initial" in Borceux's handbook (Vol.1-'2.11-p.69) but none of these terms is satisfactory. The "cofinal" name comes obviously from the vocabulary of ordered sets which are special cases, but in category theory "co" is now associated with dual notions.
There was some discussion of this point on the categories mailing list a year or two back. I think there was general consensus that the "co" in "cofinal" was redundant, and that such functors should simply be called "final". This is the term used in Mac Lane's book (section IX 3, p.217) -- I believe Mac Lane was the first to shorten "cofinal" to "final". For some reason, Borceux chose to use the opposite convention regarding "initial" and "final" in his book (although, in Exercise 2.17.8 on page 94, he seems to have reverted to the same convention as Mac Lane). Peter Johnstone