$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.2.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. The following conditions are equivalent:
- $(1)$
The diagonal map $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}})$ is right cofinal.
- $(2)$
For every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is weakly contractible.
Proof.
By virtue of Remark 7.2.3.2, condition $(1)$ is equivalent to the requirement that for 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 weakly contractible. The equivalence of $(1)$ and $(2)$ now follows from Theorem 4.6.4.17.
$\square$