Corollary 3.5.1.31. Let $f: X \rightarrow Y$ be a Kan fibration of simplicial sets. Then $f$ is a weak homotopy equivalence if and only if the relative diagonal $\delta _{X/Y}: X \rightarrow X \times _{Y} X$ is a weak homotopy equivalence and the map of connected components $\pi _0(X) \rightarrow \pi _0(Y)$ is a surjection.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$