Remark:

If $P$ is a push-out for the pair $j_1:C\longrightarrow A_1$ and $j_2:C\longrightarrow A_2$ the commutative diagram

$\begin{CD} C @> j_1 >> A_1 \\ @V{j_2}VV @VV{f_1}V\\ A_2 @> f_2 >> P \\ \end{CD}$

is also known as a cocartesian square.