Remark 3.4.1.10. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [d]^{w'} \ar [r] & X_0 \ar [d]^{w} \\ X'_{01} \ar [r] \ar [d] & X'_0 \ar [d] \\ X_1 \ar [r] & X, } \]
where $w$ and $w'$ are weak homotopy equivalences. Then the lower half of the diagram is a homotopy pullback square if and only if the outer rectangle is a homotopy pullback square (see Corollary 3.4.1.12 for a slight generalization).