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Lemma 4.31.10. Let

\[ \mathcal{A} \to \mathcal{B} \leftarrow \mathcal{C} \leftarrow \mathcal{D} \]

be a diagram of categories and functors. Then there is a canonical isomorphism

\[ \mathcal{A} \times _\mathcal {B} \mathcal{C} \times _\mathcal {C} \mathcal{D} \cong \mathcal{A} \times _\mathcal {B} \mathcal{D} \]

of categories.

Proof. Omitted. $\square$

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