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Definition 6.2.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. We say that a morphism $u: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$ if $Y$ belongs to $\operatorname{\mathcal{C}}'$ and, for every object $Z \in \operatorname{\mathcal{C}}'$, the precomposition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ [u] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. We say that $u$ exhibits $X$ as a $\operatorname{\mathcal{C}}'$-colocalization of $Y$ if $X$ belongs to $\operatorname{\mathcal{C}}'$ and, for every object $W \in \operatorname{\mathcal{C}}'$, the postcomposition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X) \xrightarrow { [u] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y)$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

We say that a subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is reflective if it is full and, for every object $X \in \operatorname{\mathcal{C}}$, there exists a morphism $u: X \rightarrow Y$ which exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$. We say that the subcategory $\operatorname{\mathcal{C}}'$ is coreflective if if is full and, for every object $Y \in \operatorname{\mathcal{C}}$, there exists a morphism $u: X \rightarrow Y$ which exhibits $X$ as a $\operatorname{\mathcal{C}}'$-coreflection of $Y$.