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