# Kerodon

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Exercise 3.4.4.1. Suppose we are given a commutative diagram of sets

$\xymatrix@R =50pt@C=50pt{ \overline{C} \ar [d] & \overline{A} \ar [l] \ar [r]^-{\overline{i}} \ar [d] & \overline{B} \ar [d] \\ C & A \ar [l] \ar [r]^-{i} & B }$

where both squares are pullback diagrams, and $i$ is a monomorphism (so that $\overline{i}$ is also a monomorphism). Show that both squares in the resulting 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 pullback squares.