Kerodon

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

Corollary 7.6.3.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ admits pullbacks if and only if, for each object $X \in \operatorname{\mathcal{C}}$, the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ admits finite products.

Proof. By virtue of Proposition 7.6.3.14, the $\infty $-category $\operatorname{\mathcal{C}}$ admits pullbacks if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ admits pairwise products. Since $\operatorname{\mathcal{C}}_{/X}$ has an initial object (given by the identity morphism $\operatorname{id}_{X}: X \rightarrow X$; see Proposition 4.6.7.22), this is equivalent to the requirement that $\operatorname{\mathcal{C}}_{/X}$ admits finite products (Corollary 7.6.1.21). $\square$