Variant 8.4.2.4. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\kappa $-small. Then the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is a dense functor.
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Proof of Variant 8.4.2.4. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\kappa $-small. We wish to show that the covariant Yoneda embedding $h^{\operatorname{\mathcal{C}}}_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is dense. Choose a cardinal $\lambda \geq \kappa $ for which the $\infty $-category $\operatorname{\mathcal{D}}= \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is locally $\lambda $-small, and let $h^{\operatorname{\mathcal{D}}}_{\bullet }: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } )$ be a covariant Yoneda embedding for $\operatorname{\mathcal{D}}$. By virtue of Proposition 8.4.1.22, it will suffice to show that the composite functor
\[ \operatorname{\mathcal{D}}\xrightarrow { h^{\operatorname{\mathcal{D}}}_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } ) \xrightarrow { \circ h^{\operatorname{\mathcal{C}}}_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } ) \]
is fully faithful. Applying Proposition 8.4.2.5, we see that this functor is isomorphic to the inclusion of $\operatorname{\mathcal{D}}= \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ as a full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } )$. $\square$