Example [efr-D7W0]

Let \mathcal {C} be any category, and let \mathcal {C}^\to denote the arrow category. Then the codomain functor \mathcal {C}^\to \to \mathcal {C} is a fibration if and only if \mathcal {C} admits all pullbacks, and in this case the functors f^*: \mathcal {C}_Y \to \mathcal {C}_X, given f: X \to Y, are given by pullback along f.

The functors f^* are sometimes referred to as pullback, a convention we generally adopt. They are also sometimes called base-change functors.