Kerodon

Remark 7.1.5.3. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $U$-initial objects. Then the restriction $U|_{\operatorname{\mathcal{C}}_0}: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}$ is fully faithful. Similarly, $U$ is fully faithful when restricted to the full subcategory of $U$-final objects of $\operatorname{\mathcal{C}}$.