Kerodon

Definition 6.2.2.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ be a functor. We will say that a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor if, for every object $X \in \operatorname{\mathcal{C}}$, the morphism $\eta _{X}: X \rightarrow L(X)$ exhibits $L(X)$ as a $\operatorname{\mathcal{C}}'$-reflection of $\operatorname{\mathcal{C}}$, in the sense of Definition 6.2.2.1. We say that a natural transformation $\epsilon : L \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}}$ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-coreflection functor if, for every object $Y \in \operatorname{\mathcal{C}}$, the morphism $\epsilon _{Y}: L(Y) \rightarrow Y$ exhibits $L(Y)$ as a $\operatorname{\mathcal{C}}'$-coreflection of $Y$.