Thanks Andr'e for pointing out by various examples why it is not always wise to insist on invariance under equivalence or weak equivalence. This in my opinion is the real issue and not whether "evil" is a tasteless name or not. The problem rather is that people using "evil" really mean it so even if they deny it. An little comment on structure versus property which is an importnat distinction in my eyes. The notion of Grothendieck fibration is a property of functors and not an additional structure. However, the notion of fibration in a(n abstract) 2-category can be formulated only postulating a certain kind of structure which, however, is unique up to canonical isomorphism. But this amounts to defining Grothendieck fibrations in terms of cleavages (which certainly are all canonically isomorphic). But choosing cleavages amounts to accepting very strong choice principles which is maybe no real problem but at least aesthetically moderately pleasing. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]