Lemma 8.8.5. Let $\mathcal{C}$ be a site. Let $\mathcal{X}$ be a fibred category over $\mathcal{C}$. The stackification of the inertia fibred category $\mathcal{I}_\mathcal {X}$ is inertia of the stackification of $\mathcal{X}$.

Proof. This follows from the fact that stackification is compatible with $2$-fibre products by Lemma 8.8.4 and the fact that there is a formula for the inertia in terms of $2$-fibre products of categories over $\mathcal{C}$, see Categories, Lemma 4.34.1. $\square$

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