I am writing about matters to do with computation of colimits of a category X in terms of colimits of a category B when there is a bifibration P: X --> B. Terminology already in use is P is cartesian P is cocartesian a lifting of u in B to \phi in X may be cartesian, cocartesian on the other hand Paul Taylor, following Peter Johnstone, I understand, uses \phi is prone, supine, instead of cartesian, cocartesian For the cofibration (?opfibration?) Ob: Groupoids --> Sets, Philip Higgins (1971) and I (1968) have previously used `universal' for cocartesian. In this situation, I would be happier with say 0-final instead of universal. But `supine' does not ring a bell with me, and carries a pejorative tone. Maybe for the general situation P: X --> B we could use P-initial, P-final morphism in X for cartesian, cocartesian morphism which would at least carry some intuition as to the meaning. Comments? I need to make a decision soon for the revision of my old topology book. Not much will be changed, and I might leave the old terminology and refer to more modern uses. However for the book on Nonabelian algebraic topology, I really do need to use modern terminologym, whatever that is, so it would be best to be consistent. I have been looking at Thomas Streicher's notes on fibrations, and at Paul Taylor's Practical Foundations. For my interest, see slides of a recent seminar at Oxford www.bangor.ac.uk/r.brown/oxford2811105.pdf called `Induced constructions and their computation'. Ronnie Brown www.bangor.ac.uk/r.brown