Remark 9.2.1.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f$ be a morphism of $\operatorname{\mathcal{C}}$ which is the colimit of a diagram
\[ K \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \quad \quad v \mapsto f_{v} \]
which is preserved by the evaluation functors $\operatorname{ev}_{0}, \operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$. 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 7.4.1.18 and 7.1.3.13.