Warning 8.4.5.4. Let $\mathbb {K}$ be a collection of simplicial sets and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. In general, it is not true that every object of $\widehat{\operatorname{\mathcal{C}}}$ can be recovered as the colimit of a diagram
\[ K \rightarrow \operatorname{\mathcal{C}}\xrightarrow {h} \widehat{\operatorname{\mathcal{C}}} \]
for some $K \in \mathbb {K}$.