Proposition 3.4.4.3. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \overline{C} \ar [d] & \overline{A} \ar [l] \ar [r]^-{\overline{i}} \ar [d] & \overline{B} \ar [d] \\ C & A \ar [r]^-{i} \ar [l] & B, } \]
in which both squares are homotopy pullbacks. If $i$ and $\overline{i}$ are monomorphisms, then both squares in the induced diagram
\[ \xymatrix@R =50pt@C=50pt{ \overline{C} \ar [r] \ar [d] & \overline{C} \coprod _{ \overline{A} } \overline{B} \ar [d] & \overline{B} \ar [l] \ar [d] \\ C \ar [r] & C \coprod _{ A} B & B \ar [l] } \]
are also also homotopy pullbacks.