Corollary 2.4.6.8. The fully faithful embedding
\[ \operatorname{Cat}\hookrightarrow \operatorname{Cat_{\Delta }}\quad \quad \operatorname{\mathcal{D}}\mapsto \underline{\operatorname{\mathcal{D}}}_{\bullet } \]
of Example 2.4.2.4 admits a left adjoint, given on objects by the formation of homotopy categories $\operatorname{\mathcal{C}}_{\bullet } \mapsto \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.