Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 6.3.3.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $w: X \rightarrow Y$ be a morphism which belongs to $W$. Then, for every $W$-local object $Z$ of $\operatorname{\mathcal{C}}$, precomposition with the homotopy class $[w]$ induces a bijection $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(Y, Z) \xrightarrow { \circ [w] } \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Z)$. In particular, if the objects $X$ and $Y$ are $W$-local, then $w$ is an isomorphism.