Kerodon

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

Proposition 7.1.5.16. Let $K$ be a simplicial set, let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits $K$-indexed limits, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty $-categories. The following conditions are equivalent:

$(1)$

The $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed limits and the functor $F$ preserves $K$-indexed limits.

$(2)$

The functor $F$ creates $K$-indexed limits.

Proof. The implication $(1) \Rightarrow (2)$ is immediately. Conversely, suppose that $(2)$ is satisfied and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Since $\operatorname{\mathcal{D}}$ admits $K$-indexed limits, $F \circ u$ can be extended to a limit diagram in $\operatorname{\mathcal{D}}$. Since $F$ creates $K$-indexed limits, it follows that there exists a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ with $\overline{u}|_{K} = u$ such that $F \circ \overline{u}$ is a limit diagram in $\operatorname{\mathcal{D}}$. Applying Remark 7.1.5.11, we see that this holds for every limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{u}|_{K} =u$, which proves $(1)$. $\square$