categories: Terminology

[Jean Benabou <jean.benabou@wanadoo.fr>]

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.
The "initial" name is even less satisfactory, because:
(i) If  Y=1, F is identified with an object  x of X and F is "initial" iff x
is a terminal object of X  !
(ii) More generally, if Y has a terminal object  t  then F is "initial" iff 
F(t) is terminal !
(iii) Even more generally yet, without assuming the existence of terminal
objects in Y or X :
 Let X^ and Y^ be the categories of presheaves on X and Y, and  F! :X^----->
Y^  the canonical extension of F to these categories.If  T is the terminal
object of Y^ one can easily show that  F has the previous property iff 
F!(T) is terminal in X^.(Which by the way, gives the nicest proof of the
stability under composition of such functors)
I propose to call these functors either "terminal" or better "final" but I
would like to know if this would not conflict with previous terminology.
Thanks for your help.