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Proposition Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $S$ be a collection of vertices of $\operatorname{\mathcal{C}}$. Then there exists a unique full simplicial subset $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ having vertex set $S$.

Proof. Take $\operatorname{\mathcal{C}}'$ to be the simplicial subset of $\operatorname{\mathcal{C}}$ consisting of those simplices $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ having the property that, for $0 \leq i \leq n$, the vertex $\sigma (i)$ belongs to $S$. $\square$