Corollary 6.2.4.7. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, let $\operatorname{\mathcal{E}}$ denote the oriented fiber product $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ (see Definition 4.6.4.1). Assume that $F_0$ is a cocartesian fibration. Then:
- $(1)$
The homotopy fiber product $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1$ is a reflective subcategory of $\operatorname{\mathcal{E}}$.
- $(2)$
Let $w$ be a morphism in the $\infty $-category $\operatorname{\mathcal{E}}$. Then $w$ is a $\operatorname{\mathcal{E}}'$-local equivalence if and only if its image in $\operatorname{\mathcal{C}}_0$ is $U$-cocartesian and its image in $\operatorname{\mathcal{C}}_1$ is an isomorphism.
Proof.
It follows from Corollary 5.3.7.3 that projection onto the second factor determines a cocartesian fibration $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}_{1}$. For each object $C \in \operatorname{\mathcal{C}}_1$, let $\operatorname{\mathcal{E}}_{C}$ denote the fiber $V^{-1} \{ C\} $, and define $\operatorname{\mathcal{E}}'_{C}$ similarly. We then have a pullback diagram of simplicial sets
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}_{C} \ar [r]^{\pi _ C} \ar [d]^{U_ C} & \operatorname{\mathcal{C}}_0 \ar [d]^{ F_0} \\ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ C\} \ar [r] & \operatorname{\mathcal{C}}. } \]
Recall that the $\infty $-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ C\} $ has a final object, and the full subcategory spanned by the final objects is the homotopy fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \{ C\} $ (Proposition 4.6.7.22). It follows that $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \{ C\} $ is a reflective subcategory of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ C\} $ (Example 6.2.2.10). Moreover, $\operatorname{\mathcal{E}}'_{C}$ is the inverse image of this reflective subcategory under the functor $U_{C}$. Our assumption that $F_0$ is a cocartesian fibration guarantees that $U_ C$ is also a cocartesian fibration. Applying Proposition 6.2.4.1, we deduce that $\operatorname{\mathcal{E}}'_{C}$ is a reflective subcategory of $\operatorname{\mathcal{E}}_{C}$. Moreover, a morphism $w$ of $\operatorname{\mathcal{E}}_{C}$ is a $\operatorname{\mathcal{E}}'_{C}$-local equivalence if and only if it is $U_{C}$-cocartesian: that is, if and only if $Q_{C}(w)$ is an $F_0$-cocartesian morphism in $\operatorname{\mathcal{C}}_0$.
Let $f: C \rightarrow D$ be a morphism in the $\infty $-category $\operatorname{\mathcal{C}}_{1}$, and let $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ be given by covariant transport along $f$ (for the cocartesian fibration $V$). Since the projection map $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}_0$ carries $V$-cocartesian morphisms to isomorphisms (Corollary 5.3.7.3), the composition $Q_{D} \circ f_{!}$ is isomorphic to $Q_{C}$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{C}}_0 )$. It follows that the functor $f_{!}$ carries $\operatorname{\mathcal{E}}'_{C}$-local equivalences in $\operatorname{\mathcal{E}}_{C}$ to $\operatorname{\mathcal{E}}'_{D}$-local equivalences in $\operatorname{\mathcal{E}}_{D}$. Assertion $(1)$ now follows from Proposition 6.2.4.2, and assertion $(2)$ from Remark 6.2.4.3.
$\square$