Kerodon

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Exercise 4.5.9.9. Let $\operatorname{\mathcal{C}}= \Delta ^2$ be the standard $2$-simplex, let $\operatorname{\mathcal{D}}= \operatorname{N}_{\bullet }( \{ 0 < 2 \} )$ be the long edge of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{E}}= \{ 0\} \coprod \{ 2\}$ be its boundary. Let $V: \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{C}}$ be the inclusion maps. Show that $U$ are $V$ are isofibrations of $\infty$-categories but that the projection map $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ can be identified with the horn inclusion $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$, which is not an inner fibration.