$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 5.1.6.3. Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^-{U} & \operatorname{\mathcal{C}}' \ar [d]^-{U'} \\ \operatorname{\mathcal{D}}\ar [r]^-{ \overline{F} } & \operatorname{\mathcal{D}}'. } \]
Assume that $U$ and $U'$ are isofibrations of $\infty $-categories and that $F$ and $\overline{F}$ are equivalences of $\infty $-categories. Then:
Proof.
We will prove the first assertion; the second follows by a similar argument. Assume first that $U'$ is a right fibration of $\infty $-categories. Then $U'$ is a cartesian fibration (Proposition 5.1.4.14), so Corollary 5.1.6.2 implies that $U$ is a cartesian fibration. To prove that $U$ is a right fibration, it will suffice to show that for every object $D \in \operatorname{\mathcal{D}}$, the fiber $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is a Kan complex (Proposition 5.1.4.14). This follows from Remark 4.5.1.21, since the functor $F$ induces an equivalence of $\infty $-categories $F_{D}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}'_{ \overline{F}(D)}$ (Corollary 4.5.2.32).
We now prove the reverse implication. Arguing as in the proof of Corollary 5.1.6.2, we can construct a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r]^-{G} \ar [d]^-{U'} & \operatorname{\mathcal{C}}\ar [d]^-{U} \\ \operatorname{\mathcal{D}}' \ar [r]^-{\overline{G}} & \operatorname{\mathcal{D}}, } \]
where $G$ and $\overline{G}$ are homotopy inverses of the equivalences $F$ and $\overline{F}$, respectively. It then follows from the preceding argument that if $U$ is a right fibration of $\infty $-categories, then $U'$ is also a right fibration of $\infty $-categories.
$\square$