Remark 4.8.4.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then, for every integer $n \geq -1$, the homotopy category $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_0)}$ can be identified with the full subcategory of $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ spanned by the images of objects which belong $\operatorname{\mathcal{C}}_0$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$