Kerodon

Warning 9.4.6.18. In the formulation of Proposition 9.4.6.17, the assumption that $U$ is an isofibration cannot be omitted. For example, let $\operatorname{\mathcal{C}}$ be a contractible Kan complex containing two vertices $C_0$ and $C_1$, let $\operatorname{\mathcal{D}}_1$ be an $\infty $-category which is the idempotent completion of a full subcategory $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}_1$, and let $\operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}_1$ be the full subcategory spanned by those objects $(C_ i, D)$ where $D$ is contained in $\operatorname{\mathcal{D}}_ i$. It follows from Example 9.4.6.10 that the projection map $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a flat inner fibration. However, if $\operatorname{\mathcal{D}}_0$ is not idempotent complete, then $U$ is not an isofibration.