Corollary 6.2.2.11. 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 morphism of simplicial sets. Let $\overline{f}: \overline{X} \rightarrow \overline{Y}$ be a morphism in the $\infty $-category $\operatorname{\mathcal{C}}_{/q}$, having image $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$. If $f$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$, then $\overline{f}$ exhibits $\overline{Y}$ as a $\operatorname{\mathcal{C}}'_{/q}$-reflection of $\overline{X}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Set $\operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}_{f/} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}'$. Our assumption that $f$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$ guarantees that the restriction map $u: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a trivial Kan fibration (Proposition 6.2.2.10). We wish to show that the analogous restriction map
\[ \overline{u}: ( \operatorname{\mathcal{C}}_{/q} )_{ \overline{f} / } \times _{ \operatorname{\mathcal{C}}_{/q} } \operatorname{\mathcal{C}}'_{/q} \rightarrow ( \operatorname{\mathcal{C}}_{/q} )_{ \overline{X} / } \times _{ \operatorname{\mathcal{C}}_{/q} } \operatorname{\mathcal{C}}'_{/q} \]
is also trivial Kan fibration. Let us regard $\overline{f}$ as a morphism of simplicial sets $\Delta ^1 \star K \rightarrow \operatorname{\mathcal{C}}$, which we can identify with a diagram $\overline{q}: K \rightarrow \operatorname{\mathcal{D}}$. Under this identification, $\overline{u}$ corresponds to the map $\operatorname{\mathcal{D}}_{ / \overline{q} } \rightarrow \operatorname{\mathcal{E}}_{ / u \circ \overline{q} }$ induced by $u$, which is a trivial Kan fibration by virtue of Corollary 4.3.7.17. $\square$