$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 9.2.4.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $W$-local objects. Assume that:
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally small and admits small colimits.
The collection $W$ is small.
For each morphism $w: X \rightarrow Y$ which belongs to $W$, the objects $X$ and $Y$ are $\kappa $-compact for some small infinite cardinal $\kappa $.
Then $\operatorname{\mathcal{C}}'$ is a reflective localization of $\operatorname{\mathcal{C}}$.
Proof.
For each morphism $w: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, choose a morphism $\delta _{w}: Y \coprod _{X} Y \rightarrow Y$ which is a relative codiagonal of $w$ (see Variant 7.6.2.16). Note that, if $X$ and $Y$ are $\kappa $-compact for some infinite cardinal $\kappa $, then the pushout $Y \coprod _{X} Y$ is also $\kappa $-compact (Proposition 
). Let $W'$ be the smallest collection of morphisms of $\operatorname{\mathcal{C}}$ which contains $W$ and is closed under the construction $w \mapsto \gamma _{w}$. By virtue of Remark 9.2.1.12, an object of $\operatorname{\mathcal{C}}$ is $W$-local if and only if it is $W'$-local. We may therefore replace $W$ by $W'$ and thereby reduce to proving Corollary 9.2.4.6 in the special case where $W$ is closed under the formation of relative codiagonals.
Fix an object $C \in \operatorname{\mathcal{C}}$. Using Theorem 9.2.4.3, we see that there exists a morphism $f: C \rightarrow C'$, where $C'$ is weakly $W$-local and $f$ is a transfinite pushout of morphisms which belong to $W$. Using Proposition 9.2.3.15, we see that $C'$ belongs to the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$. To complete the proof, it will suffice to show that $f$ exhibits $C'$ as a $\operatorname{\mathcal{C}}'$-reflection of $C$: that is, every object of $\operatorname{\mathcal{C}}'$ is $f$-local. This follows from Remark 9.2.4.2.
$\square$