Definition 4.33.5. Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a category over $\mathcal{C}$. We say $\mathcal{S}$ is a fibred category over $\mathcal{C}$ if given any $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$ lying over $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and any morphism $f : V \to U$ of $\mathcal{C}$, there exists a strongly cartesian morphism $f^*x \to x$ lying over $f$.
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