Proposition 8.4.1.8. Let $\operatorname{\mathcal{D}}$ be a locally small $\infty $-category. A full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense if and only if the restricted Yoneda embedding $\operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is fully faithful.
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Proof of Proposition 8.4.1.8. Let $\operatorname{\mathcal{D}}$ be a locally small $\infty $-category and let $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ be a full subcategory. By virtue of Example 8.4.1.16, it will suffice to show that the inclusion functor $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}$ is dense if and only if the restricted Yoneda embedding $\operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is fully faithful. Following the convention of Remark 4.7.0.5, this is a special case of Proposition 8.4.1.22. $\square$