The Stacks project

Lemma 14.21.2. Let $\mathcal{C}$ be a category. Let $U$ be an $m$-truncated simplicial object of $\mathcal{C}$. For any $n \leq m$ the colimit

\[ \mathop{\mathrm{colim}}\nolimits _{([n]/\Delta )_{\leq m}^{opp}} U(n) \]

exists and is equal to $U_ n$.

Proof. This is so because the category $([n]/\Delta )_{\leq m}$ has an initial object, namely $\text{id} : [n] \to [n]$. $\square$

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