# Kerodon

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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.