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Corollary 8.4.3.9. Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small, and let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Then every object $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ can be realized as the colimit of a diagram

\[ \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}\xrightarrow {h_{\bullet }} \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ), \]

where $\operatorname{\mathcal{K}}$ is a $\kappa $-small $\infty $-category.

Proof. Let $\operatorname{\mathcal{K}}'$ denote the fiber product $ \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\kappa })} \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F}}$. Combining Lemma 8.4.3.5 with Theorem 8.3.3.13, we deduce that $\operatorname{\mathcal{K}}'$ is essentially $\kappa $-small. We can therefore choose an equivalence of $\infty $-categories $e: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{K}}'$, where $\operatorname{\mathcal{K}}$ is $\kappa $-small. Applying Theorem 8.4.3.3, we deduce that $\mathscr {F}$ is a colimit of the composite functor

\[ \operatorname{\mathcal{K}}\xrightarrow {e} \operatorname{\mathcal{K}}' = \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\kappa })} \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F}} \rightarrow \operatorname{\mathcal{C}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ). \]
$\square$