Exercise 6.2.2.23. Suppose that the conditions of Corollary 6.2.2.22 are satisfied and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory of $\operatorname{\mathcal{C}}$. Show that $\eta $ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor if and only if the following conditions are satisfied:
For each object $X \in \operatorname{\mathcal{C}}$, the object $L(X)$ is contained in $\operatorname{\mathcal{C}}'$.
For each object $Y \in \operatorname{\mathcal{C}}'$, there exists an isomorphism $Y \rightarrow L(X)$ for some object $X \in \operatorname{\mathcal{C}}$.
If the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is replete (Example 4.4.1.12), then it is uniquely determined by these conditions.