Corollary 7.1.2.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. The following conditions are equivalent:
- $(1)$
The diagram $u$ has a limit in $\operatorname{\mathcal{C}}$.
- $(2)$
The oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ u\} \rightarrow \operatorname{\mathcal{C}}$ has a final object.
- $(3)$
The slice $\infty $-category $\operatorname{\mathcal{C}}_{/u}$ has a final object.