What I need is a structure which resembles a 2-category, but in which the _union_ of all homsets is a category. The apparent localization of 1-categories to constant (single) homsets comes from looking *only* at the vertical compositions, which have a stationary or static character with respect to endpoints. Use the horizontal compositions as well as the vertical when you need to move the endpoints of homsets around. Lax comma categories (if I recall correctly these are in John Gray's 1974 or so LNM book on 2-categories) give a further generalization of this when you need to restrict or otherwise circumscribe the domain of variation of the endpoints in order to make your category. But this is just icing on top of the essential machinery provided by 2-categories. Vaughan Pratt