$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 8.4.3.5. Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small, and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor. Then the $\infty $-category
\[ \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}} = \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )} \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}} \]
is essentially $\kappa $-small.
Proof.
The $\infty $-category $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is equivalent to $\operatorname{\mathcal{C}}$ (Theorem 8.3.3.13), and is therefore essentially $\kappa $-small. Since $\kappa $ is regular, it will suffice to show that each fiber of the right fibration $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}} \rightarrow \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is an essentially $\kappa $-small Kan complex (Corollary 5.6.7.7). Equivalently, we must show that for each object $\mathscr {G} \in \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$, the mapping space $X = \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })}( \mathscr {G}, \mathscr {F} )$ is essentially $\kappa $-small. This follows from Proposition 8.3.1.3: if $\mathscr {G}$ is representable by the object $C \in \operatorname{\mathcal{C}}$, then $X$ is homotopy equivalent to the $\kappa $-small Kan complex $\mathscr {F}(C)$.
$\square$