Let C be a category and I and I' graphs (or categories if you prefer). Define a morphism of diagrams psi:(delta:I-->C)-->(delta':I'-->C) to be a graph morphism (or functor if you prefer) psi:I-->I' together with a natural transformation alpha:delta' o psi-->delta. This definition turns Lim into a contravariant functor from the category of diagrams to C (when C is complete, anyway). I believe this construction has been familiar since the early days of category theory, but I don't know a reference and would be glad to learn of any. By the way, Barr in SLN 236 (page 52) defines an entirely different notion of morphism of diagrams which Tholen and Tozzi develop extensively in "Completions of Categories and Initial Completions", Cahiers 1989, pages 127-156. This makes Lim a covariant functor. Charles Wells, 105 South Cedar Street, Oberlin, Ohio 44074, USA. (I am on sabbatical until 20 August 1997 and cannot easily be reached at Case Western Reserve University.) EMAIL: cfw2@po.cwru.edu. HOME PHONE: 216 774 1926. FAX: Same as home phone. HOME PAGE: URL http://www.cwru.edu/CWRU/Dept/Artsci/math/wells/home.html "Some have said that I can't sing. But no one will say that I _didn't_ sing." --Florence Foster Jenkins