Dear Jean, of course, you are right when emphasizing that one need choice for classes to endow an "anonymous" fibration with a cleavage. But that applies also to catgeories with say binary products. One needs choice for classes in order to choose a product cone for every pair of objects. In many instances, however, categories come together with a choice of products and fibrations come together with a choice of a cleavage. For example Set comes with a choice of a cleavage. Fibrations arising from internal categories are even split. Many constructions on fibrations allow one to choose a cleavage given cleavages for the arguments. Do you know of any construction on fibrations which is not "cleavage preserving" in this sense? Of course, one should not require cartesian functors to preserve cleavages just as one should not require functors to preserve chosen products. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]