Kerodon

Construction 8.4.5.5. Let $\mathbb {K}$ be a collection of simplicial sets and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Choose an uncountable regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small and every simplicial set $K \in \mathbb {K}$ is essentially $\kappa $-small. We let $\widehat{\operatorname{\mathcal{C}}}$ denote the smallest full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ which contains all representable functors and is closed under the formation of $K$-indexed colimits, for each $K \in \mathbb {K}$. Note that covariant Yoneda embedding for $\operatorname{\mathcal{C}}$ determines a functor $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$, which is dense (by virtue of Variant 8.4.2.4 and Remark 8.4.1.19) and fully faithful (by virtue of Theorem 8.3.3.13).