Charles Wells asked the following: __________ 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. ______________ Steve Lack replied with the folowing information: ____________ The dual construction (i.e. for colimits) appears in Rene Guitart, ``Remarques sur les machines et les structures'', Cahiers XV-2 (1974); and its sequel Rene Guitart and Luc Van den Bril, ``Decompositions et lax-completions'', Cahiers XVIII-4 (1977); where further references are also given. _____________ I am writing at the university, with my files at home; but my memory is that the construction was introduced by Eilenberg and Mac Lane in 1945, in a paper called something like "On a general theory of natural equivalences". Max Kelly.