Corollary 7.1.4.23 (Colimits in a Reflective Localization). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory (Definition 6.2.2.1), and let $u: K \rightarrow \operatorname{\mathcal{C}}'$ be a diagram. If $u$ admits a colimit in $\operatorname{\mathcal{C}}$, then it also admits a colimit in $\operatorname{\mathcal{C}}'$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Proposition 6.2.2.13, the inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$. If $u$ admits a colimit in $\operatorname{\mathcal{C}}$, then $L \circ u$ admits a colimit in $\operatorname{\mathcal{C}}'$ (Corollary 7.1.4.22). Since $u$ factors through $\operatorname{\mathcal{C}}'$, it is isomorphic to $L \circ u$ and therefore also admits a colimit in $\operatorname{\mathcal{C}}'$ (Remark 7.1.1.8). $\square$