Corollary 6.2.2.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $q: K \rightarrow \operatorname{\mathcal{C}}'$ be a diagram. If $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$, then $\operatorname{\mathcal{C}}'_{/q}$ is a reflective subcategory of $\operatorname{\mathcal{C}}_{/q}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\overline{X}$ be an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/q}$; we wish to show that there exists a morphism $\overline{f}: \overline{X} \rightarrow \overline{Y}$ which exhibits $\overline{Y}$ as a $\operatorname{\mathcal{C}}'_{/q}$-reflection of $\overline{X}$. Let $X$ denote the image of $\overline{X}$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$, we can choose a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ which exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$. By virtue of Corollary 6.2.2.11, it will suffice to show that $f$ can be lifted to a morphism $\overline{f}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{C}}_{/q}$. Unwinding the definitions, we can rewrite this as a lifting problem
\[ \xymatrix { \emptyset \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}' \ar [d] \\ K \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}', } \]
which admits a solution by virtue of the fact that the right vertical map is a trivial Kan fibration (Proposition 6.2.2.10). $\square$