Example 3.4.1.13. Suppose we are given a commutative diagram of simplicial sets
3.44
\begin{equation} \begin{gathered}\label{equation:homotopy-pullback-contractible-corner} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^{f_0} \\ X_1 \ar [r]^-{f_1} & X } \end{gathered} \end{equation}
where $X$ is weakly contractible. Then (3.44) is a homotopy pullback square if and only if the induced map $X_{01} \rightarrow X_0 \times X_1$ is a weak homotopy equivalence. To prove this, we can use Corollary 3.4.1.12 to reduce to the case $X = \Delta ^0$, in which case it follows from the criterion of Corollary 3.4.1.6.