Remark 8.4.5.5. 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, where $\operatorname{\mathcal{D}}$ is $\mathbb {K}$-cocomplete. It follows from Proposition 8.4.5.3 that there exists a functor $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. In this case, there exists a $\mathbb {K}$-cocontinuous functor $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ and an isomorphism of functors $\alpha : F \rightarrow f \circ h$, as indicated in the diagram
Moreover, the pair $( F, \alpha )$ is unique up to isomorphism. In this situation, we say that $\alpha $ exhibits $F$ as the $\mathbb {K}$-cocontinuous extension of $f$.