Remark 9.5.7.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f$ be a morphism of $\operatorname{\mathcal{C}}$ which is the levelwise colimit of a diagram
\[ K \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \quad \quad v \mapsto f_{v}. \]
If an object $C \in \operatorname{\mathcal{C}}$ is $f_{v}$-local for each vertex $v \in K$, then it is also $f$-local. This follows from Propositions 7.4.1.22 and 7.1.3.18.