Lemma 4.32.5. Let $\mathcal{C}$ be a category. Let $f : \mathcal{X} \to \mathcal{S}$ and $g : \mathcal{Y} \to \mathcal{S}$ be morphisms of categories over $\mathcal{C}$. For any object $U$ of $\mathcal{C}$ we have the following identity of fibre categories

\[ \left(\mathcal{X} \times _\mathcal {S}\mathcal{Y}\right)_ U = \mathcal{X}_ U \times _{\mathcal{S}_ U} \mathcal{Y}_ U \]

**Proof.**
Omitted.
$\square$

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