Kerodon

Variant 9.2.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $w: X \rightarrow Y$ and $w': X' \rightarrow Y'$ be morphisms of $\operatorname{\mathcal{C}}$, and suppose that $w'$ is a retract of $w$ (in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$). If an object $C \in \operatorname{\mathcal{C}}$ is weakly $w$-local, then it is also weakly $w'$-local. In particular, if we regard the object $C \in \operatorname{\mathcal{C}}$ is fixed, then the condition that $C$ is $w$-local depends only on the isomorphism class of $w$ (as an object of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$).