Definition 6.2.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. We say that a morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ is a $\operatorname{\mathcal{C}}'$-local equivalence if, for every object $Z \in \operatorname{\mathcal{C}}'$, precomposition with $u$ induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ [u] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$