The Stacks project

Lemma 4.35.14. Let $\mathcal{A} \to \mathcal{B} \to \mathcal{C}$ be functors between categories. If $\mathcal{A}$ is fibred in groupoids over $\mathcal{B}$ and $\mathcal{B}$ is fibred in groupoids over $\mathcal{C}$, then $\mathcal{A}$ is fibred in groupoids over $\mathcal{C}$.

Proof. One can prove this directly from the definition. However, we will argue using the criterion of Lemma 4.35.2. By Lemma 4.33.12 we see that $\mathcal{A}$ is fibred over $\mathcal{C}$. To finish the proof we show that the fibre category $\mathcal{A}_ U$ is a groupoid for $U$ in $\mathcal{C}$. Namely, if $x \to y$ is a morphism of $\mathcal{A}_ U$, then its image in $\mathcal{B}$ is an isomorphism as $\mathcal{B}_ U$ is a groupoid. But then $x \to y$ is an isomorphism, for example by Lemma 4.33.2 and the fact that every morphism of $\mathcal{A}$ is strongly $\mathcal{B}$-cartesian (see Lemma 4.35.2). $\square$

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