Remark 6.2.2.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, so that we can identify $\operatorname{\mathcal{C}}'^{\operatorname{op}}$ with a full subcategory of the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Then:
A morphism $u: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$ if and only if $u^{\operatorname{op}}: Y^{\operatorname{op}} \rightarrow X^{\operatorname{op}}$ exhibits $Y^{\operatorname{op}}$ as a $\operatorname{\mathcal{C}}'^{\operatorname{op}}$-coreflection of $X^{\operatorname{op}}$.
The subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is reflective if and only if the subcategory $\operatorname{\mathcal{C}}'^{\operatorname{op}} \subseteq \operatorname{\mathcal{C}}^{\operatorname{op}}$ is coreflective.