Theorem 9.4.6.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of simplicial sets. Then, for every Morita equivalence of simplicial sets $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, the projection map $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}$ is also a Morita equivalence.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Theorem 9.4.6.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of simplicial sets and let $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ be a Morita equivalence; we wish to show that the projection map $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}$ is also a Morita equivalence. Choose an uncountable regular cardinal $\kappa $ such that $U$ is essentially $\kappa $-small. Using Theorem 9.4.4.1, we can choose a diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \Delta ^1 & } \]
which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa $-cocompletion of $\operatorname{\mathcal{E}}$. By virtue of Lemma 9.4.6.12, it will suffice to show that $\widehat{U}$ is a cartesian fibration, which is a reformulation of our assumption that $U$ is flat (Proposition 9.4.6.9). $\square$