Proposition 6.2.4.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of $\infty $-categories and let $\operatorname{\mathcal{E}}' \subseteq \operatorname{\mathcal{E}}$ be a full subcategory which satisfies the following conditions:
- $(a)$
For each object $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}'_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$ is a reflective subcategory of $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.
- $(b)$
For each morphism $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$, the contravariant transport functor $f^{\ast }: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ carries $\operatorname{\mathcal{E}}'_{D}$ into $\operatorname{\mathcal{E}}'_{C}$.
Then $\operatorname{\mathcal{E}}'$ is a reflective subcategory of $\operatorname{\mathcal{E}}$.
Proof of Proposition 6.2.4.4.
Let $X$ be an object of $\operatorname{\mathcal{E}}$ having image $C = U(X)$ in $\operatorname{\mathcal{C}}$. It follows from $(a)$ that there exists a $\operatorname{\mathcal{E}}'_{C}$-local equivalence $w: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$, where $Y$ belongs to $\operatorname{\mathcal{E}}'_{C}$. We will complete the proof by showing that $w$ is a $\operatorname{\mathcal{E}}'$-local equivalence (and therefore exhibits $Y$ as a $\operatorname{\mathcal{E}}'$-localization of $X$). Fix an object $Z \in \operatorname{\mathcal{E}}'$ having image $D = U(Z)$ in $\operatorname{\mathcal{C}}$; we wish to show that precomposition with $w$ induces a homotopy equivalence $\theta : \operatorname{Hom}_{\operatorname{\mathcal{E}}}( Y, Z ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z)$. As in the proof of Proposition 6.2.4.2, it suffices to prove this in the special case where $\operatorname{\mathcal{C}}= \Delta ^1$, with $C = 0$ and $D = 1$.
Invoking assumption $(b)$, we can choose a $U$-cartesian morphism $s: Z' \rightarrow Z$, where $Z' \in \operatorname{\mathcal{E}}'_{C}$. We then have a homotopy commutative diagram of Kan complexes
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}_{C}}(Y,Z') \ar [d]^{ \circ [w] } \ar [r]^{ [s] \circ } & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(Y,Z) \ar [d]^{ \circ [w] } \\ \operatorname{Hom}_{\operatorname{\mathcal{E}}_{C}}(X,Z') \ar [r]^{ [s] \circ } & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z). } \]
Since $s$ is $U$-cartesian, the horizontal maps are homotopy equivalences (Corollary 5.1.2.3). Consequently, to show that $\theta $ is a homotopy equivalence, it will suffice to show that the left vertical map is a homotopy equivalence. This follows immediately from our assumption that $w$ is a $\operatorname{\mathcal{E}}'_{C}$-local equivalence in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$.
$\square$