Example 5.3.8.3. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category equipped with a functor $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^{n}$. Then $U$ is automatically an inner fibration (Proposition 4.1.1.10), which is minimal (in the sense of Definition 5.3.8.1) if and only if the $\infty $-category $\operatorname{\mathcal{E}}$ is minimal (in the sense of Definition 4.7.6.4). The “only if” direction is immediate from the definitions, and the converse is a special case of Example 5.3.8.2.
In particular, an $\infty $-category $\operatorname{\mathcal{E}}$ minimal if and only if the projection map $\operatorname{\mathcal{E}}\rightarrow \Delta ^{0}$ is a minimal inner fibration.