Warning 3.4.1.8. For a general pair of morphisms $f_0: X_0 \rightarrow X$, $f_1: X_1 \rightarrow X$ in the category of simplicial sets, there need not exist a homotopy pullback square
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^{f_0} \\ X_1 \ar [r]^-{f_1} & X. } \]
For example, if $f_0: \{ 0\} \hookrightarrow \Delta ^1$ and $f_1: \{ 1\} \hookrightarrow \Delta ^1$ are the inclusion maps, then the existence of a commutative diagram
3.42
\begin{equation} \label{diagram:homotopy-pullback-square4} \begin{gathered} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & \{ 0\} \ar [d]^{f_0} \\ \{ 1\} \ar [r]^-{f_1} & \Delta ^1 } \end{gathered}\end{equation}
guarantees that the simplicial set $X_{01}$ is empty, in which case (3.42) is not a homotopy pullback square.