Proposition 8.4.5.7. Let $\mathbb {K}$ be a collection of simplicial sets, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and define $\widehat{\operatorname{\mathcal{C}}}$ as in Construction 8.4.5.5. Then the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$.
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Proof of Proposition 8.4.5.7. Let $\mathbb {K}$ be a collection of simplicial sets, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $\widehat{\operatorname{\mathcal{C}}}$ be as in Construction 8.4.5.5. By construction, the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is $\mathbb {K}$-cocomplete. To complete the proof, we must show that if $\operatorname{\mathcal{D}}$ is any $\mathbb {K}$-cocomplete $\infty $-category, then composition with the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ induces an equivalence of $\infty $-categories $\theta : \operatorname{Fun}^{\mathbb {K}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
Let $\operatorname{\mathcal{C}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ be the essential image of $h_{\bullet }$, so that $\theta $ factors as a composition
\[ \operatorname{Fun}^{\mathbb {K}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \xrightarrow {\theta '} \operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{D}}) \xrightarrow {\theta ''} \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \]
where $\theta ''$ is an equivalence of $\infty $-categories (Theorem 8.3.3.13). Using Lemma 8.4.5.9, we see that $\operatorname{Fun}^{\mathbb {K}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ is the full subcategory of $\operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ spanned by those functors which are left Kan extended from $\operatorname{\mathcal{C}}'$. It follows from Corollary 7.3.6.15 that $\theta '$ is a trivial Kan fibration onto a full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{D}})$; in particular, it is fully faithful, so that $\theta $ is fully faithful. Lemma 8.4.5.8 implies that $\theta $ is essentially surjective, and therefore an equivalence of $\infty $-categories (Theorem 4.6.2.20). $\square$