Kerodon

Remark 6.3.1.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to a collection of edges $W$. Let $[W]$ denote the collection of morphisms in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ which belong to the image of $W$. Then the induced functor $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ exhibits the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ as a $1$-categorical localization of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ with respect to $[W]$, in the sense of Definition 6.3.0.6. This follows immediately from Example 6.3.1.6.