Warning 3.4.0.8. The conclusion of Proposition 3.4.0.7 is generally false if neither $f_0$ or $f_1$ is assumed to be a Kan fibration. For example, suppose that $X$ is a Kan complex containing vertices $x$ and $y$. If $x \neq y$, then the fiber product $\{ x\} \times _{X} \{ y\} $ is empty. However, the homotopy fiber product $\{ x\} \times ^{\mathrm{h}}_{X} \{ y\} $ is not necessarily empty: its vertices can be identified with edges $p: x \rightarrow y$ having source $x$ and target $y$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$