Kerodon

Remark 4.5.1.3. Let $\mathbf{Cat}$ denote the (strict) $2$-category of categories (Example 2.2.0.4) and let $\mathrm{h} \mathit{\operatorname{Cat}}$ denote its homotopy category (Construction 2.2.8.12). Then the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ determines a fully faithful functor from $\mathrm{h} \mathit{\operatorname{Cat}}$ to the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$ of Construction 4.5.1.1. This functor admits a left adjoint, which carries an $\infty $-category $\operatorname{\mathcal{C}}$ to its homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.