Proof.
The equivalence $(1) \Leftrightarrow (2)$ follows from Example 5.3.8.3 and the implication $(3) \Rightarrow (1)$ is a special case of Proposition 5.3.8.7. We will complete the proof by showing that $(1)$ implies $(3)$. Assume that condition $(1)$ is satisfied. For $0 \leq i \leq n$, let $\operatorname{\mathcal{E}}_{i}$ denote the fiber $U^{-1} \{ i\} = \{ i\} \times _{ \Delta ^ n } \operatorname{\mathcal{E}}$. For $1 \leq i \leq n$, choose a functor $T_{i}: \operatorname{\mathcal{E}}_{i-1} \rightarrow \operatorname{\mathcal{E}}_ i$ which is given by covariant transport for the cocartesian fibration $U$. Let $\mathscr {F}: [n] \rightarrow \operatorname{Set_{\Delta }}$ be the diagram given by the sequence of functors
\[ \operatorname{\mathcal{E}}_0 \xrightarrow {T_1} \operatorname{\mathcal{E}}_1 \xrightarrow { T_2 } \operatorname{\mathcal{E}}_{2} \xrightarrow {T_3} \cdots \xrightarrow {T_ n} \operatorname{\mathcal{E}}_ n. \]
Set $\operatorname{\mathcal{E}}' = \operatorname{N}_{\bullet }^{\mathscr {F} }([n])$, so that we have a canonical isomorphism
\[ F_0: \operatorname{sk}_{0}( \Delta ^ n ) \times _{ \Delta ^ n } \operatorname{\mathcal{E}}\xrightarrow {\sim } \operatorname{sk}_0( \Delta ^ n ) \times _{ \Delta ^ n } \operatorname{\mathcal{E}}'. \]
Let $\operatorname{Spine}[n]$ denote the spine of the standard $n$-simplex $\Delta ^ n$ (Example 1.5.7.7). Since the functors $T_ i$ are given by covariant transport with respect to $U$, Remark 5.2.4.3 guarantees that we can extend $F_0$ to a morphism
\[ F_1: \operatorname{Spine}[n] \times _{ \Delta ^ n } \operatorname{\mathcal{E}}\xrightarrow {\sim } \operatorname{Spine}[n] \times _{ \Delta ^ n } \operatorname{\mathcal{E}}' \]
which is an equivalence of cocartesian fibrations over $\operatorname{Spine}[n]$. Since the inclusion map $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne (Example 1.5.7.7), Lemma 5.3.6.5 guarantees that the inclusion $\operatorname{Spine}[n] \times _{\Delta ^ n} \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}$ is a categorical equivalence of simplicial sets, so that $F_1$ extends to a functor $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$. We then have a commutative diagram
\[ \xymatrix { \operatorname{Spine}[n] \times _{ \Delta ^ n } \operatorname{\mathcal{E}}\ar [d]^{F_1} \ar [r] & \operatorname{\mathcal{E}}\ar [d]^{F} \\ \operatorname{Spine}[n] \times _{\Delta ^ n} \operatorname{\mathcal{E}}' \ar [r] & \operatorname{\mathcal{E}}', } \]
where the horizontal maps are categorical equivalences by Lemma 5.3.6.5 and the left vertical map is a categorical equivalence by Proposition 5.1.7.5. It follows that $F$ is an equivalence of $\infty $-categories. The $\infty $-category $\operatorname{\mathcal{E}}$ is minimal by assumption, and the $\infty $-category $\operatorname{\mathcal{E}}'$ is minimal by virtue of Proposition 5.3.8.7. Applying Proposition 4.7.6.13, we conclude that $F$ is an isomorphism of simplicial sets.
$\square$