Theorem 6.3.5.1. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^{F} & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \operatorname{\mathcal{C}}\ar [r]^{\overline{F}} & \operatorname{\mathcal{C}}' } \]
which satisfies the following conditions:
- $(1)$
The morphisms $U$ and $U'$ are cocartesian fibrations.
- $(2)$
The morphism $F$ carries $U$-cocartesian edges of $\operatorname{\mathcal{E}}$ to $U'$-cocartesian edges of $\operatorname{\mathcal{E}}'$.
- $(3)$
For every vertex $C \in \operatorname{\mathcal{C}}$ having image $C' = \overline{F}(C) \in \operatorname{\mathcal{C}}'$, the induced functor of $\infty $-categories $F_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C'}$ exhibits $\operatorname{\mathcal{E}}'_{C'}$ as the localization of $\operatorname{\mathcal{E}}_{C}$ with respect to some collection of morphisms $W_{C}$ of $\operatorname{\mathcal{E}}_{C}$.
- $(4)$
The morphism $\overline{F}$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to some collection of morphisms $\overline{W}$ of $\operatorname{\mathcal{C}}$.
Set $W_{-} = \bigcup _{C \in \operatorname{\mathcal{C}}} W_{C}$ and let $W_{+}$ be the collection of all $U$-cocartesian edges $e$ such that $U(e)$ belongs to $\overline{W}$. Then $F$ exhibits $\operatorname{\mathcal{E}}'$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W_{-} \cup W_{+}$.
Proof of Theorem 6.3.5.1.
Suppose we are given a commutative diagram of simplicial sets
6.14
\begin{equation} \begin{gathered}\label{equation:fiberwise-localization} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^{F} & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \operatorname{\mathcal{C}}\ar [r]^{\overline{F}} & \operatorname{\mathcal{C}}' } \end{gathered} \end{equation}
which satisfies the hypotheses of Theorem 6.3.5.1. Fix an $\infty $-category $\operatorname{\mathcal{D}}$. We wish to show that precomposition with $F$ induces a fully faithful functor $F^{\ast }: \operatorname{Fun}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{D}})$, whose essential image consists of those morphisms $G: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ which carry each edge of $W_{-} \cup W_{+}$ to an isomorphism in $\operatorname{\mathcal{D}}$. Let $\pi : \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{C}}'} \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}'$ be given by projection onto the second factor. Note that the pair $(U, F)$ determines a morphism of simplicial sets $\widetilde{F}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{C}}'} \operatorname{\mathcal{E}}'$ satisfying $\pi \circ \widetilde{F} = F$, so that $F^{\ast }$ factors as a composition
\[ \operatorname{Fun}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{D}}) \xrightarrow {\pi ^{\ast }} \operatorname{Fun}( \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{C}}'} \operatorname{\mathcal{E}}', \operatorname{\mathcal{D}}) \xrightarrow {\widetilde{F}^{\ast }} \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{D}}). \]
Let $W'$ be the collection of all edges of $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{C}}'} \operatorname{\mathcal{E}}'$ of the form $( \overline{e}, f)$, where $\overline{e}$ belongs to $\overline{W}$ and $f$ is a $U'$-cocartesian edge of $\operatorname{\mathcal{E}}'$. It follows from Proposition 6.3.5.4 that the functor $\pi ^{\ast }$ is fully faithful, and that its essential image consists of those morphisms $G': \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{C}}'} \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{D}}$ which carry each edge of $W'$ to an isomorphism in $\operatorname{\mathcal{D}}$. Applying Proposition 6.3.5.2, we see that the functor $\widetilde{F}^{\ast }$ is also fully faithful, and that its essential image consists of those morphisms $G: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ which carry each edge of $W_{-}$ to an isomorphism in $\operatorname{\mathcal{D}}$. To complete the proof, it will suffice to show the following:
- $(\ast )$
A morphism of simplicial sets $G': \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{C}}'} \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{D}}$ carries each edge of $W'$ to an isomorphism in $\operatorname{\mathcal{D}}$ if and only if $G' \circ \widetilde{F}$ carries each edge of $W_{+}$ to an isomorphism in $\operatorname{\mathcal{D}}$.
The “only if” assertion is immediate (since $\widetilde{F}(W_{+} )$ is contained in $W'$). The converse follows from the observation that every edge $(\overline{e}, f)$ is isomorphic, when viewed as an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \Delta ^1, \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{C}}'} \operatorname{\mathcal{E}}')$, to $\widetilde{F}(e)$, where $e: X \rightarrow Y$ is any $U$-cocartesian edge of $\operatorname{\mathcal{E}}$ for which $U(e) = \overline{e}$ and $F(X)$ is isomorphic to the domain of $f$ as an object of the $\infty $-category $\{ X\} \times _{\operatorname{\mathcal{C}}'} \operatorname{\mathcal{E}}'$.
$\square$