Proposition 7.7.3.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ is locally cartesian closed if and only if it admits pullbacks and, for every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the pullback functor $f^{\ast }: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}_{/X}$ admits a right adjoint.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Remark 7.7.3.15, we may assume that $\operatorname{\mathcal{C}}$ admits pullbacks. Then, for every object $Y \in \operatorname{\mathcal{C}}$, the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$ admits finite limits (Example 7.6.2.31). Applying Proposition 7.7.3.13, we conclude that $\operatorname{\mathcal{C}}_{/Y}$ is cartesian closed if and only if, for every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the pullback functor $f^{\ast }: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}_{/X}$ admits a right adjoint. Proposition 7.7.3.19 now follows by allowing the object $Y$ to vary. $\square$