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Definition 8.4.5.1. Let $\mathbb {K}$ be a collection of simplicial sets. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is $\mathbb {K}$-cocomplete if it admits $K$-indexed colimits, for each $K \in \mathbb {K}$. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\mathbb {K}$-cocomplete $\infty $-categories, we let $\operatorname{Fun}^{\mathbb {K}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors which preserve $K$-indexed colimits, for each $K \in \mathbb {K}$. We say that a functor of $\infty $-categories $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$ if the following conditions are satisfied:

  • The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is $\mathbb {K}$-cocomplete.

  • For every $\mathbb {K}$-cocomplete $\infty $-category $\operatorname{\mathcal{D}}$, precomposition with $h$ induces an equivalence of $\infty $-categories $\operatorname{Fun}^{\mathbb {K}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.