Example 4.8.4.3. Suppose that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are Kan complexes. In this case, the equivalence $(1) \Leftrightarrow (4)$ of Theorem 4.8.4.1 reduces to the combinatorial version of Whitehead's theorem: a morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a homotopy equivalence if and only if it is bijective on connected components and induces an isomorphism of homotopy groups (Theorem 3.2.7.1). See Proposition 4.8.2.20.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$