Corollary 6.2.2.22 (02FG)

Corollary 6.2.2.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $L$ be a functor from $\operatorname{\mathcal{C}}$ to itself, and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ be a natural transformation. The following conditions are equivalent:

$(1)$: For every object $X \in \operatorname{\mathcal{C}}$, the morphisms $L(\eta _ X): L(X) \rightarrow L(L(X))$ and $\eta _{L(X)}: L(X) \rightarrow L(L(X))$ are isomorphisms.
$(2)$: There exists a full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ for which $\eta $ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor, in the sense of Definition 6.2.2.14.

Proof. The implication $(2) \Rightarrow (1)$ follows from Proposition 6.2.2.17. Conversely, suppose that condition $(1)$ is satisfied, and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by those objects of the form $L(X)$ for $X \in \operatorname{\mathcal{C}}$. Assumption $(1)$ guarantees that $\eta _{Y}$ is an isomorphism for each $Y \in \operatorname{\mathcal{C}}'$, so that $\eta $ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor by virtue of Proposition 6.2.2.17. $\square$