Lemma 4.33.3 (09WU)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 4: Categories
Section 4.33: Fibred categories
Lemma 4.33.3 (cite)

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Lemma 4.33.3. Let $F : \mathcal{A} \to \mathcal{B}$ and $G : \mathcal{B} \to \mathcal{C}$ be composable functors between categories. Let $x \to y$ be a morphism of $\mathcal{A}$ . If $x \to y$ is strongly $\mathcal{B}$ -cartesian and $F(x) \to F(y)$ is strongly $\mathcal{C}$ -cartesian, then $x \to y$ is strongly $\mathcal{C}$ -cartesian.

Proof. This follows directly from the definition. $\square$

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