Proposition 4.5.2.10. A commutative diagram of Kan complexes
4.22
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square5} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^{q} \\ X_1 \ar [r] & X } \end{gathered} \end{equation}
is a categorical pullback square if and only if it is a homotopy pullback square.