Corollary 9.2.5.18. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which admits a left adjoint. Then $F$ is $\kappa $-left exact for every regular cardinal $\kappa $.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. For every object $D \in \operatorname{\mathcal{D}}$, our assumption guarantees the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ has an initial object (given by $F(D)$; see Corollary 6.2.6.2). In particular, $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is $\kappa $-cofiltered (Example 9.1.1.6). Allowing $D$ to vary, we conclude that $F$ is $\kappa $-left exact (Theorem 9.2.5.15). $\square$