Lemma 4.35.12. Let $\mathcal{C}$ be a category. If $p : \mathcal{S} \to \mathcal{C}$ is fibred in groupoids, then so is the inertia fibred category $\mathcal{I}_\mathcal {S} \to \mathcal{C}$.

Proof. Clear from the construction in Lemma 4.34.1 or by using (from the same lemma) that $I_\mathcal {S} \to \mathcal{S} \times _{\Delta , \mathcal{S} \times _\mathcal {C} \mathcal{S}, \Delta }\mathcal{S}$ is an equivalence and appealing to Lemma 4.35.7. $\square$

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