Kerodon

Remark 8.5.6.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Evaluation on the unique vertex of $\Delta ^1 / \operatorname{\partial \Delta }^1$ induces an isofibration of $\infty $-categories $\operatorname{End}_{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$. Moreover, for each object $X \in \operatorname{\mathcal{C}}$, the fiber $\{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{End}_{\operatorname{\mathcal{C}}}$ can be identified with the endomorphism space $\operatorname{End}_{\operatorname{\mathcal{C}}}(X) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)$ of Variant 4.6.1.3.