Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 6.2.2.7. In the situation of Definition 6.2.2.6, the assumption that $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor guarantees in particular that for every object $X \in \operatorname{\mathcal{C}}$, the image $L(X)$ belongs to the full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$. Consequently, we can also view $L$ as a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{C}}'$.