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Proposition 6.3.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory, and let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a left adjoint to the inclusion functor $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$. Let $W$ be the collection of all morphisms $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ for which $L(f)$ is an isomorphism in $\operatorname{\mathcal{C}}'$. Then:

$(1)$

The collection $W$ is localizing (Definition 6.3.3.2).

$(2)$

Every object of $\operatorname{\mathcal{C}}'$ is $W$-local (Definition 6.3.3.1).

$(3)$

If $\operatorname{\mathcal{C}}'$ is replete, then every $W$-local object of $\operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{C}}'$.

Proof. We first prove $(2)$. Let $Z$ be an object of $\operatorname{\mathcal{C}}'$ and let $w: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which belongs to $W$; we wish to show that precomposition with the homotopy class $[w]$ induces an isomorphism

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, Z) \xrightarrow { \circ [w] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. Using Proposition 6.2.1.17, we can identify $\theta $ with the map

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( L(Y), Z) \xrightarrow { \circ [L(w)] } \operatorname{Hom}{\operatorname{\mathcal{C}}'}( L(X), Z), \]

which is invertible by virtue of our assumption that $L(w)$ is an isomorphism of $\operatorname{\mathcal{C}}'$.

We now prove $(1)$. It follows immediately from the definitions that $W$ contains all isomorphisms of $\operatorname{\mathcal{C}}$ and satisfies the two-out-of-three property. Let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow \iota \circ L$ be the unit of an adjunction. Then $\eta $ carries each object $X \in \operatorname{\mathcal{C}}$ to a morphism $\eta _ X: X \rightarrow L(X)$, where $L(X)$ belongs to $\operatorname{\mathcal{C}}'$ and is therefore $W$-local (by virtue of $(2)$). Moreover, $L(\eta _ X)$ is an isomorphism in $\operatorname{\mathcal{C}}'$ (Proposition 6.2.2.11), so $\eta _ X$ belongs to $W$.

We now prove $(3)$. Suppose that $X$ is a $W$-local object of $\operatorname{\mathcal{C}}$. Then $\eta _ X: X \rightarrow L(X)$ is a morphism between $W$-local objects of $\operatorname{\mathcal{C}}$. Since $\eta _ X$ belongs to $W$, it follows that $\eta _ X$ is an isomorphism (Remark 6.3.3.4). If the full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is replete, we conclude that $X$ belongs to $\operatorname{\mathcal{C}}'$. $\square$