Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Exercise 3.4.3.1. Suppose we are given a pushout square of sets

\[ \xymatrix { A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D. } \]

Then, for every map of sets $\overline{D} \rightarrow D$, the induced diagram

\[ \xymatrix { A \times _{D} \overline{D} \ar [r] \ar [d] & B \times _{D} \overline{D} \ar [d] \\ C \times _{D} \overline{D} \ar [r] & \overline{D} } \]

is also a pushout square.