If $s:\mathcal{X} \to \mathcal{C}$ is a fibered category and $\varphi:C \to D$ is a morphism in $\mathcal{C}$, for each object $y$ in the fiber $\mathcal{X}_D$ the axiom of choice allows us to specify exactly one pullback $f:y_C \to y$ (i.e., a cartesian arrow $f$ with $s(f)=\varphi$). The choice of such a collection is called a 'cleavage', and a cleavage always exists by the axiom of choice.
This enables us to define the 'change of base functor' $\varphi^*:\mathcal{X}_D \to \mathcal{X}_C$.
