The Stacks project

Lemma 4.42.4. Let $\mathcal{C}$ be a category. Let $\mathcal{X}$, $\mathcal{Y}$ be categories fibred in groupoids over $\mathcal{C}$. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism. If $F$ is representable then every one of the functors

\[ F_ U : \mathcal{X}_ U \longrightarrow \mathcal{Y}_ U \]

between fibre categories is faithful.

Proof. Clear from the description of fibre categories in Lemma 4.42.1 and the characterization of representable fibred categories in Lemma 4.40.2. $\square$

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