Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 7.1.4.3. 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.