Exercise 3.4.3.1. Suppose we are given a pushout square of sets
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D. } \]
Then, for every map of sets $\overline{D} \rightarrow D$, the induced diagram
\[ \xymatrix@R =50pt@C=50pt{ A \times _{D} \overline{D} \ar [r] \ar [d] & B \times _{D} \overline{D} \ar [d] \\ C \times _{D} \overline{D} \ar [r] & \overline{D} } \]
is also a pushout square.