Terminology re fibrations and opfibrations of categories

["Ronald Brown" <ronnie@ll319dg.fsnet.co.uk>]

To add to my previous email, I'd like reactions to the following
terminology:

Let P: X \to B be a functor. A morphism u: x \to y in X is  cofinal w.r.t.
P, and y is the P-final object w,r,t u and P , if ... (and here we have the
usual notion of cocartesian).

Dually,  u is coinitial, and x is the initial object w.r.t   u and P if ...
(and here we have the usual notion of cartesian).

In situations where P is understood, we can then talk about cofinal and
coinitial morphisms, and structures or objects or (in my case, groupoids).

An advantage is that the direction of the notion and its dual should be
clear.

If f=P(u), I would then write \bar{f}: x \to f_*(x) in the first case, and
\underline{f}: f^*(y) \to y in the second. I would also call f_*(x) the
object induced by f.  What is a handy name for f^*(y)? The restriction of y
by f?

All these notions occur for modules, crossed modules, ...... and relate to
change of base.

Ronnie
www.bangor.ac.uk/r.brown