Remark 4.1.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a subcategory, let $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be an $n$-simplex of $\operatorname{\mathcal{C}}$ for $n > 0$. The following conditions are equivalent:
- $(1)$
The $n$-simplex $\sigma $ is contained in the subcategory $\operatorname{\mathcal{C}}'$.
- $(2)$
For every pair of integers $0 \leq i < j \leq n$, the edge
\[ \Delta ^{1} \simeq \operatorname{N}_{\bullet }( \{ i < j \} ) \hookrightarrow \Delta ^{n} \xrightarrow {\sigma } \operatorname{\mathcal{C}} \]is contained in the subcategory $\operatorname{\mathcal{C}}'$.
- $(3)$
For every integer $1 \leq j \leq n$, the edge
\[ \Delta ^{1} \simeq \operatorname{N}_{\bullet }( \{ j-1 < j \} ) \hookrightarrow \Delta ^{n} \xrightarrow {\sigma } \operatorname{\mathcal{C}} \]is contained in the subcategory $\operatorname{\mathcal{C}}'$.
The implications $(1) \Rightarrow (2) \Rightarrow (3)$ are immediate from the definitions, and the implication $(3) \Rightarrow (1)$ follows from fact that the inclusion $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ is weakly right orthogonal to the inner anodyne morphism $\operatorname{Spine}[n] \hookrightarrow \Delta ^{n}$ (see Example 1.5.7.7 and Proposition 4.1.3.1).