 A category formula_1 with a functor to a category formula_2 is called a fibered category over formula_2 if for any morphism formula_4 in formula_2 and any object formula_6 of formula_1 with image formula_8 (under the functor), there is a pullback formula_9 of formula_6 by formula_11.This means a morphism with image formula_11 such that any morphism formula_13 with image formula_14 can be factored as formula_15 by a unique morphism formula_16 in formula_1 such that the functor maps formula_18 to formula_19.The element formula_20 is called the pullback of formula_6 along formula_11 and is unique up to canonical isomorphism.