Kerodon

Example 5.3.8.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories. Suppose that $\operatorname{\mathcal{E}}$ is minimal. Then, for every $n$-simplex $\sigma : \Delta ^ m \rightarrow \operatorname{\mathcal{C}}$, the fiber product $\operatorname{\mathcal{E}}_{\sigma } = \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a simplicial subset of the minimal $\infty $-category $\Delta ^ n \times \operatorname{\mathcal{E}}$ (Remark 4.7.6.9), and is therefore also minimal (Remark 4.7.6.8). It follows that $U$ is a minimal inner fibration.