The Stacks project

Lemma 14.8.2. If $U, V, W$ are simplicial objects in the category $\mathcal{C}$, and if $a : U \to V$, $b : U \to W$ are morphisms and if $V\amalg _ U W$ exists, then we have

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

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

Proof. Omitted. $\square$

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