Remark 7.2.4.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is filtered.
- $(2)$
For every finite simplicial set $K$ and every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is nonempty.
- $(3)$
For every finite simplicial set $K$ and every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, the oriented fiber product $\{ f\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$ is nonempty.
- $(4)$
For every finite simplicial set $K$ and every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, there exists a morphism $f \rightarrow f'$ in the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$, where $f': K \rightarrow \operatorname{\mathcal{C}}$ is a constant diagram.
The equivalences $(1) \Leftrightarrow (2)$ and $(3) \Leftrightarrow (4)$ follow immediately from the definitions, and the equivalence $(2) \Leftrightarrow (3)$ follows from Theorem 4.6.4.17.