As Steve Lack and Peter Johnstone said, the functors to which Jean Benabou referred,
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.
are called "final" in Saunders Mac Lane's famous book. As Peter also said, there was a lengthy discussion on "categories" in July 1998 on this topic, which you can look up in the archive at ftp://tac.mta.ca/pub/categories/1998/98-7 The footnote that I wrote on p389 of "Practical Foundations" http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/html/s73.html summing up this discussion reads
The prefix ``co-'' in the original word cofinal carried the usual Latin--English meaning of ``together,'' rather than the meaning of dualisation inherited from (co)homology (and maybe trigonometry before that).
Although final functors are the analogue, not the dual, of cofinal monotone functions, the prefix was dropped in [Mac Lane, p. 213] as it was considered inappropriate.
I feel that it was unnecessary to introduce this confusion, as Proposition 3.2.10 associates them with \emph{co}limits (but cf Exercise 3.32).
Even so, the definitions are not the same: any surjective function between discrete posets is cofinal and they give rise to the same joins, but to different coproducts. This difference is attributable to the hidden existential quantifier mentioned in the footnote on page 129.
Paul