The Stacks project

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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