# Kerodon

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Proposition 6.2.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ be the inclusion map. Then $\iota$ admits a left adjoint if and only if $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$. Similarly, $\iota$ admits a right adjoint if and only if $\operatorname{\mathcal{C}}'$ is a coreflective subcategory of $\operatorname{\mathcal{C}}$.

Proof of Proposition 6.2.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. It follows from Proposition 6.2.2.11 that the inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint if and only if there exists a functor $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ and a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ which exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor. By virtue of Lemma 6.2.2.10, this is equivalent to the requirement that $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$. The analogous characterization of coreflective subcategories follows by a similar argument. $\square$