Lemma 14.19.3. Let $\mathcal{C}$ be a category. Let $U$ be an $m$-truncated simplicial object of $\mathcal{C}$. For $n \leq m$ the limit $\mathop{\mathrm{lim}}\nolimits _{(\Delta /[n])_{\leq m}^{opp}} U(n)$ exists and is canonically isomorphic to $U_ n$.

**Proof.**
This is true because the category $(\Delta /[n])_{\leq m}$ has an final object in this case, namely the identity map $[n] \to [n]$.
$\square$

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