Lemma 14.21.4. If $U$ is an $m$-truncated simplicial set and $n > m$ then all $n$-simplices of $i_{m!}U$ are degenerate.

**Proof.**
This can be seen from the construction of $i_{m!}U$ in Lemma 14.21.1, but we can also argue directly as follows. Write $V = i_{m!}U$. Let $V' \subset V$ be the simplicial subset with $V'_ i = V_ i$ for $i \leq m$ and all $i$ simplices degenerate for $i > m$, see Lemma 14.18.4. By the adjunction formula, since $\text{sk}_ m V' = U$, there is an inverse to the injection $V' \to V$. Hence $V' = V$.
$\square$

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