I'm looking for language advice on dealing with the following "St. Ives" problem, as a 2-categorical down-sizing of the 4-categorical original. I have a 2-category CAT^{\pm} (arising from a 2-monad C\pm1 analogous to the monad X+1) that is disconnected in the sense that it is representable as a sum of connected 2-categories. Each summand has homcats that are similarly disconnected, being representable as a sum of connected homcats. If I take a summand 2-category, and from each of its homcats coherently take a summand 1-category, I end up with a 2-category with an initial and a final category serving as respectively the syntax and the semantics of a certain 2-theory (of linearly distributive but not self-dual categories of dually typed objects, but that's another story I'm hoping Myles Tierney will like better than the Chu story). I don't think it's caused by my doing anything wrong--the situation appears to have arisen naturally and I've found no way of making it go away. Has this situation arisen before? If so, what is the proposed terminology for this kind of finality and its associated uniqueness? If not, I'm happy to make up my own conventions, but I don't want to reinvent the wheel here. Vaughan Pratt