Lemma 4.42.2 (02Y6)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 4: Categories
Section 4.42: Representable 1-morphisms
Lemma 4.42.2 (cite)

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Lemma 4.42.2. 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. Let $G : \mathcal{C}/U \to \mathcal{Y}$ be a $1$ -morphism. Then

$(\mathcal{C}/U) \times _\mathcal {Y} \mathcal{X} \longrightarrow \mathcal{C}/U$

is a category fibred in groupoids.

Proof. We have already seen in Lemma 4.35.7 that the composition

$(\mathcal{C}/U) \times _\mathcal {Y} \mathcal{X} \longrightarrow \mathcal{C}/U \longrightarrow \mathcal{C}$

is a category fibred in groupoids. Then the lemma follows from Lemma 4.35.13. $\square$

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