Remark 9.1.2.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq 0$ be a nonnegative integer. Condition $(\ast '_{n})$ of Lemma 9.1.2.12 is equivalent to the assertion that, for every morphism of simplicial sets $f: \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{C}}$, the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is nonempty. By virtue of Theorem 4.6.4.17, this is equivalent to the requirement that the oriented fiber product $\{ f\} \operatorname{\vec{\times }}_{ \operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$ is nonempty. We can therefore reformulate $(\ast '_{n})$ as follows:
- $(\ast '_ n)$
For every diagram $f: \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{C}}$, there exists an object $C \in \operatorname{\mathcal{C}}$ and a natural transformation $f \rightarrow \underline{C}$, where $\underline{C}: \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{C}}$ is the constant morphism taking the value $C$.