With the help of Mike Shulman I have eventually understood how one arrives at a notion of "weak fibration". There is the following characterisation of P : XX -> BB being a fibration due to J. Gray : for every X in XX the functor P/X : XX/X -> BB/P(X) has a right adjoint right inverse. Of course "right inverse" is "evil" so let's replace it by the "non-evil" requirement that all P/X have a right adjoint which is full and faithful (replacing "counit is an identity" by "counit is an isomorphisms"). Working this out one sees that P is a weak fibration (i.e. a fibration in this weaker "non-evil" sense) iff for all X in XX and u : J -> P(X) there is a cartesian arrow phi : Y -> X with P(phi) isomorphic to u in BB/P(X). Just writing out the definition of "cartesian" one observes that it doesn't make reference to equality of objects. Thus, replacing "equal" by "isomorphic" in "for all u : J -> P(X) there is a cartesian arrow phi : Y - X with P(phi) equal to u in BB/P(X)" one obtains the above definition of weak fibration. So one can hardly deny that this is the right(eous) non-evil version of Grothendieck fibration. But for such weak fibrations one looses the important property that for every u : J -> I one can transport X over I to u^*X over J along u. This property is essential for category theory over an arbitrary base (topos). In other words whereas "evil" fibrations correspond to indexed categories BB^op -> Cat the weak ones do not. Moreover, indexed categories, i.e. pseudofunctors BB^op -> Cat, can be formulated in a "non-evil" way but one has to accept the bureaucracy of coherence conditions which does not show up when working with fibrations (and one also has to accept very big categories like Cat). Thus sticking to a "non-evil" discipline one comes to the conclusion that indexed categories are better than fibered categories. The latter are more elegant and easier to work with (one need not sweep under the carpet coherence issues all the time). This suggest to me that a better name for "non-evil" might be "puritan". This suggestion of "puritan" is meant as seriously as the suggestion of "evil". Thomas PS As pointed out by Andre Joyal when reasoning about the most "non-evil" category of homotopy types it seems to be essential to use concepts which are not stable under weak equivalence. Another instance of the principle "good" requires "evil". [For admin and other information see: http://www.mta.ca/~cat-dist/ ]