Kerodon

Corollary 8.4.6.10. Let $\mathbb {K}$ be a collection of simplicial sets and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. Let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be the smallest full subcategory which contains the essential image of $f_0: f|_{\operatorname{\mathcal{C}}_0}$ and is closed under the formation of $K$-indexed colimits, for each $K \in \mathbb {K}$. Then the functor $f_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}_0$ exhibits $\operatorname{\mathcal{D}}_0$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}_0$.