Proof of uniqueness of a push-out

Question

I'm reading the note about push-outs, and there's this proposition:

What is that universal property of Z that the author's talking about? I have no idea what it is so the rest of the proof doesn't make sense to me.

Answer 1

Given a diagram $$ \require{AMScd} \begin{CD} A @>{f}>> Y\\ @V{g}VV \\ X, \end{CD} $$ a push-out is an object $Z$ together with morphisms $F\colon X\to Z$ and $G\colon Y\to Z$ such that the diagram $$ \begin{CD} A @>{f}>> Y\\ @V{g}VV @VV{G}V\\ X @>{F}>> Z \end{CD} $$ commutes and the following universal property is satisfied:

Given any other $Z'$ with morphisms $F'$ and $G'$ such that $$ \begin{CD} A @>{f}>> Y\\ @V{g}VV @VV{G'}V\\ X @>{F'}>> Z' \end{CD} $$ commutes, there is a unique morphism $\psi\colon Z\to Z'$ such that $F'=\psi\circ F$ and $G'=\psi\circ G$: