Lemma 14.21.9. Let $\mathcal{A}$ be an abelian category Let $U$ be an $m$-truncated simplicial object of $\mathcal{A}$. For $n > m$ we have $N(i_{m!}U)_ n = 0$.

Proof. Write $V = i_{m!}U$. Let $V' \subset V$ be the simplicial subobject of $V$ with $V'_ i = V_ i$ for $i \leq m$ and $N(V'_ i) = 0$ for $i > m$, see Lemma 14.18.9. 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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