Example 7.6.4.12. Let $X$ be a Kan complex containing vertices $x$ and $y$. Then there exists an equalizer diagram
7.71
\begin{equation} \begin{gathered}\label{equation:bad-equalizer} \xymatrix@C =50pt@R=50pt{ \{ x\} \times ^{\mathrm{h}}_{X} \{ y\} \ar [r]^-{f} & \Delta ^0 \ar@ <.4ex>[r]^-{x} \ar@ <-.4ex>[r]_-{y} & X } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{S}}$ (for a more general statement, see Corollary 7.6.4.21). However, unless the homotopy fiber product $\{ x\} \times ^{\mathrm{h}}_{X} \{ y\} $ is either empty or contractible, the image of (7.71) in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ is not an equalizer diagram (since the homotopy class $[f]$ is a monomorphism in $\mathrm{h} \mathit{\operatorname{Kan}}$).