I have seen very often the following "abstract" definition of a fibration in a 2-category C : A map (i.e. a 1-cell) p: X --> S is a fibration iff for each object Y of C the functor C(Y,p): C(Y,X) --> C(Y,S) is a fibration (in the usual sense) which depends "2-functorially" on Y. Such an "obvious" definition is much too naive and does not give the correct notion in most examples. 1- Even if C= Cat, the 2-category of (small) categories, a fibration in the abstract sense is a Grothendieck fibration which admits a cleavage. Thus if we don't assume AC, which we don't need to define fibrations, it does not coincide with the usual one. 2- The situation is much worse in more general cases. Suppose E is a topos (this assumption is much too strong), and take C = Cat(E), the category of internal categories in E. On can define internal fibrations, and "fibrations" in the previous "abstract" sense. They do not coincide. It all boils down to the following remark: E and (E°, Set) are Toposes, the Yoneda functor E --> (E*,Set) preserves an reflects limits, but "nothing else" of the internal logic, which is needed to define internal fibrations. Best to all, Jean [For admin and other information see: http://www.mta.ca/~cat-dist/ ]