The Stacks project

Lemma 14.10.2. If $U, V, W$ are cosimplicial objects in the category $\mathcal{C}$, and if $a : V \to U$, $b : W \to U$ are morphisms and if $V \times _ U W$ exists, then we have

\[ \mathop{\mathrm{Mor}}\nolimits (T, V \times _ U W) = \mathop{\mathrm{Mor}}\nolimits (T, V) \times _{\mathop{\mathrm{Mor}}\nolimits (T, U)} \mathop{\mathrm{Mor}}\nolimits (T, W) \]

for any fourth cosimplicial object $T$ of $\mathcal{C}$.

Proof. Omitted. $\square$

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