Corollary 11.5.0.11 (01VC)

Corollary 11.5.0.11. Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^-{q} & \operatorname{\mathcal{C}}' \ar [d]^-{q'} \\ \operatorname{\mathcal{D}}\ar [r]^-{\overline{F}} & \operatorname{\mathcal{D}}'. } \]

Assume that:

$(1)$: The functors $q$ and $q'$ are isofibrations.
$(2)$: The isofibration $q$ is locally cartesian and the functor $F$ carries locally $q$-cartesian morphisms of $\operatorname{\mathcal{C}}$ to locally $q'$-cartesian morphisms of $\operatorname{\mathcal{C}}'$.
$(3)$: The functor $\overline{F}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}'$ is an equivalence of $\infty $-categories.

Then $F$ is an equivalence of $\infty $-categories if and only if, for every object $D \in \operatorname{\mathcal{D}}$ having image $D' = \overline{F}(D) \in \operatorname{\mathcal{D}}'$, the induced map of fibers $F_{D}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}'_{D'}$ is an equivalence of $\infty $-categories. Moreover, if this condition is satisfied, then $q'$ is also a locally cartesian fibration.

Proof. If $F$ is an equivalence of $\infty $-categories, then Corollary 4.5.2.32 guarantees that each $F_{D}$ is an equivalence of $\infty $-categories. The converse follows by combining Proposition 5.1.6.7. $\square$