Corollary 9.2.5.21. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $\kappa $ be a regular cardinal, and let $G: K \rightarrow \operatorname{\mathcal{C}}$ be a $\kappa $-small diagram. If $F$ is $\kappa $-left exact, then the induced map of slice $\infty $-categories $F_{/G}: \operatorname{\mathcal{C}}_{/G} \rightarrow \operatorname{\mathcal{D}}_{/(F \circ G)}$ is also $\kappa $-left exact.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Theorem 9.2.5.15, it will suffice to show that for each object $\widetilde{D} \in \operatorname{\mathcal{D}}_{/ (F \circ G)}$, the $\infty $-category
\[ \operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}_{/G} \times _{ \operatorname{\mathcal{D}}_{ / F \circ G} } (\operatorname{\mathcal{D}}_{ / F \circ G})_{ \widetilde{D}/ } \]
is $\kappa $-cofiltered. Let $D$ denote the image of $\widetilde{D}$ in $\operatorname{\mathcal{D}}$, and set $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$. Then $\widetilde{D}$ determines a lift of $G$ to a diagram $\widetilde{G}: K \rightarrow \widetilde{\operatorname{\mathcal{C}}}$, and $\operatorname{\mathcal{E}}$ can be identified with the slice $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}_{ / \widetilde{G} }$. By Proposition 9.1.1.14, it will suffice to show that the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$ is $\kappa $-cofiltered, which follows from our assumption that $F$ is $\kappa $-left exact (Theorem 9.2.5.15). $\square$