Kerodon

Remark 9.2.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which is closed under transfinite composition. Then $W$ is closed under isomorphism: that is, if $f$ and $g$ are morphisms of $\operatorname{\mathcal{C}}$ which are isomorphic (as objects of the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$), then $f$ belongs to $W$ if and only if $g$ belongs to $W$. This follows by combining Example 9.2.2.6 with Remark 9.2.2.8. In particular, the condition that a morphism $f$ of $\operatorname{\mathcal{C}}$ belongs to $W$ depends only on the homotopy class $[f]$.