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Lemma 4.31.13. Let $\mathcal{A} \to \mathcal{C}$, $\mathcal{B} \to \mathcal{C}$ and $\mathcal{C} \to \mathcal{D}$ be functors between categories. Then the diagram

\[ \xymatrix{ \mathcal{A} \times _\mathcal {C} \mathcal{B} \ar[d] \ar[r] & \mathcal{A} \times _\mathcal {D} \mathcal{B} \ar[d] \\ \mathcal{C} \ar[r]^-{\Delta _{\mathcal{C}/\mathcal{D}}} \ar[r] & \mathcal{C} \times _\mathcal {D} \mathcal{C} } \]

is a $2$-fibre product diagram.

Proof. Omitted. $\square$

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