Kerodon

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

Corollary 7.1.3.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: A \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $K$ be an arbitrary simplicial set. Then:

$(1)$

If $\operatorname{\mathcal{C}}$ admits $K$-indexed limits, then the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ admits $K$-indexed limits and the projection map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}$ preserves $K$-indexed limits.

$(2)$

If $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits, then the slice $\infty $-category $\operatorname{\mathcal{C}}_{/f}$ admits $K$-indexed colimits and the projection map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ preserves $K$-indexed colimits.