$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 8.4.8.7. Let $\mathbb {K}$ be a collection of simplicial sets, let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ be $\infty $-categories which are $\mathbb {K}$-cocomplete, and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a right fibration which is $\mathbb {K}$-cocontinuous. Let $X \in \operatorname{\mathcal{E}}$ be an object having image $C = U(X)$. Fix an uncountable regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ are locally $\kappa $-small and each $K \in \mathbb {K}$ is $\kappa $-small, so that $C$ and $X$ determine corepresentable functors
\[ h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \kappa } \quad \quad h^{X}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}_{< \kappa }. \]
If $h^{C}$ is $\mathbb {K}$-cocontinuous, then $h^{X}$ is $\mathbb {K}$-cocontinuous.
Proof.
Suppose we are given a colimit diagram $\mathscr {F}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}$ for some $K \in \mathbb {K}$; we wish to show that $h^{X} \circ \mathscr {F}$ is a colimit diagram in $\operatorname{\mathcal{S}}_{< \kappa }$. By virtue of Corollary 7.4.3.14, it will suffice to show that the upper horizontal map in the pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{X/} \times _{\operatorname{\mathcal{E}}} K \ar [r] \ar [d] & \operatorname{\mathcal{E}}_{X/} \times _{\operatorname{\mathcal{E}}} K^{\triangleright } \ar [d] \\ \operatorname{\mathcal{C}}_{C / } \times _{\operatorname{\mathcal{C}}} K \ar [r] & \operatorname{\mathcal{C}}_{C / } \times _{\operatorname{\mathcal{C}}} K^{\triangleright } } \]
is a weak homotopy equivalence of simplicial sets. Since $U$ is a right fibration, the vertical maps appearing in the diagram are Kan fibrations (Corollary 4.3.7.3). It will therefore suffice to show that the bottom horizontal map is a weak homotopy equivalence (Corollary 3.3.7.4). This follows from Corollary 7.4.3.14, together with our assumption that $h^{C}$ preserves the colimit diagram $U \circ \mathscr {F}$.
$\square$