Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 6.3.1.15. 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 $W$. Then, for every Kan complex $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a homotopy equivalence of Kan complexes

\[ \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) = \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]

(see Example 6.3.1.4). It follows that $F$ is a weak homotopy equivalence of simplicial sets.