Kerodon

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Corollary 4.4.1.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the tautological map $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ is an isofibration of $\infty $-categories.

Proof. It follows from Proposition 4.1.1.10 that $U$ is an inner fibration. Since $U$ induces an isomorphism of homotopy categories, Proposition 4.4.1.7 guarantees that $U$ is an isofibration. $\square$