Remark 9.2.4.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which is a transfinite pushout of morphisms which belong to $W$. Then:
If an object $C \in \operatorname{\mathcal{C}}$ is weakly $W$-local, then it is weakly $f$-local.
If an object $C \in \operatorname{\mathcal{C}}$ is $W$-local, then it is $f$-local.
The first assertion follows from Propositions 9.2.3.14 and 9.2.3.17; the second follows from Remark 9.2.1.10 and Variant 9.2.3.18.