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Proposition 6.2.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(a)$

Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects. Then $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$ and $W$ is the collection of $\operatorname{\mathcal{C}}'$-local equivalences.

$(b)$

There exists a reflective subcategory $\operatorname{\mathcal{C}}'' \subseteq \operatorname{\mathcal{C}}$ such that $W$ is the collection of $\operatorname{\mathcal{C}}''$-local equivalences.

$(c)$

The collection of morphisms $W$ is localizing (Definition 6.2.3.9).

$(d)$

The collection of morphisms $W$ satisfies conditions $(1)$, $(2_{-})$, and $(3)$ of Lemma 6.2.3.11.

Proof. The implications $(a) \Rightarrow (b)$ and $(c) \Rightarrow (d)$ are trivial, the implication $(b) \Rightarrow (c)$ follows from Example 6.2.3.10, and the implication $(d) \Rightarrow (a)$ follows from Lemma 6.2.3.11. $\square$