Variant 9.2.3.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $W$ be the collection of morphisms $w: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ such that $C$ is $w$-local. Then $W$ is closed under transfinite composition.
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Proof. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $W$ be the collection of morphisms $w$ of $\operatorname{\mathcal{C}}$ such that $C$ is $w$-local. Then $W$ contains all identity morphisms, is closed under composition (Remark 9.2.1.11), and is closed under the formation of colimits in the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ which are preserved by the evaluation functors $\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ (Remark 9.2.1.9). Applying Proposition 9.2.2.10 (and Remark 9.2.2.11), we conclude that $W$ is closed under transfinite composition. $\square$