Proposition 9.4.6.29. Let $U_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}_0$ be a flat isofibration of simplicial sets. Then there exists a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [r] \ar [d]^{U_0} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}_0 \ar [r]^-{\iota } & \operatorname{\mathcal{C}}, } \]
where $U$ is a flat isofibration of $\infty $-categories. Moreover, we may assume that $\iota $ is inner anodyne.
Proof.
Using Corollary 9.4.6.26, we can choose an inner anodyne map $\iota : \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ and a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [r] \ar [d]^{U_0} & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \operatorname{\mathcal{C}}_0 \ar [r]^-{\iota } & \operatorname{\mathcal{C}}, } \]
where $U'$ is a flat isofibration. Using Proposition 9.4.6.22, we can factor $U'$ as a composition $\operatorname{\mathcal{E}}' \xrightarrow {F} \operatorname{\mathcal{E}}'' \xrightarrow {U''} \operatorname{\mathcal{C}}$, where $F$ is an equivalence of $\infty $-categories and $U''$ is a flat isofibration. Set $\operatorname{\mathcal{E}}''_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}''$, so that we have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [r] \ar [d]^{F_0} & \operatorname{\mathcal{E}}' \ar [d]^{F} \\ \operatorname{\mathcal{E}}''_0 \ar [r] & \operatorname{\mathcal{E}}''. } \]
Since $U'$ and $U''$ are flat, Proposition 9.4.6.6 guarantees that the horizontal maps are categorical equivalences. Since $F$ is an equivalence of $\infty $-categories, it follows that $F_0$ is a categorical equivalence of simplicial sets (Remark 4.5.3.5), and therefore an equivalence of isofibrations over $\operatorname{\mathcal{C}}_0$ (Proposition 5.1.7.5). Applying Lemma 5.6.7.1, we conclude that there exists a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [d] \ar [r] & \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{G} & \operatorname{\mathcal{E}}'' \ar [d]^{U''} \\ \operatorname{\mathcal{C}}_0 \ar [r]^-{\iota } & \operatorname{\mathcal{C}}\ar [r]^-{\operatorname{id}} & \operatorname{\mathcal{C}}, } \]
where the left square is a pullback, the upper horizontal composition is $F_0$, and $G$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$. Since $U''$ is a flat isofibration, it follows that $U$ is also a flat isofibration (Remark 9.4.6.3 and Proposition 5.1.7.14).
$\square$