Definition 4.33.6. Assume $p : \mathcal{S} \to \mathcal{C}$ is a fibred category.

A

*choice of pullbacks*^{1}for $p : \mathcal{S} \to \mathcal{C}$ is given by a choice of a strongly cartesian morphism $f^\ast x \to x$ lying over $f$ for any morphism $f: V \to U$ of $\mathcal{C}$ and any $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$.Given a choice of pullbacks, for any morphism $f : V \to U$ of $\mathcal{C}$ the functor $f^* : \mathcal{S}_ U \to \mathcal{S}_ V$ described above is called a

*pullback functor*(associated to the choices $f^*x \to x$ made above).

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