Notation 1.1.0.13. Let $S_{\bullet }$ be a simplicial set and let $\sigma \in S_{n}$ be an $n$-simplex of $\operatorname{\mathcal{C}}$. By virtue of Proposition 1.1.0.12, there is a unique morphism $f_{\sigma }: \Delta ^ n \rightarrow S_{\bullet }$ in the category of simplicial sets which satisfies $f_{\sigma }( \operatorname{id}_{[n]} ) = \sigma $. In practice, we will often abuse notation by identifying the $n$-simplex $\sigma $ with the morphism $f_{\sigma }$.
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