What is the definition of pushout of a short exact sequence?
In this paper1, page $126$, under the proof of Proposition $2.8$, I don't understand how the author justify the existence of the operator $V:Z_X \rightarrow Z$.
The following is the two exact sequences from the paper. (Assume we don't have the operator $V$ now)
The author proved that $S\overline{L}(Z_X)=\{ 0 \}$. Then he said that there exists an operator $V : Z_X \rightarrow Z$ such that the diagram commutes.
Can anyone justify why $V$ exists?
1G. Godefroy and N. J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (2003), no. 1, 121–141. MR 2030906 (2004m:46027), DOI: 10.4064/sm159-1-6
Because the sequences are exact we can think of $Z$ as a subspace of $Y$ by identifying $Z$ with the image of $R$. And similiarly we can think of $Z_X$ as a subspace of $\mathcal{F}(X)$. Then $V$ is just the restriction of $\overline{L}$ to $Z_X$, where the condition that $S\overline{L}(Z_X)={0}$ tells us that this restriction ends up with all values in $\ker(S)$ which is $Z$ by exactness.