Functoriality. The assignment $\webleft (X,Y,Z,f,g\webright )\mapsto X\mathbin {\textstyle \coprod _{f,Z,g}}Y$ defines a functor
\[ -_{1}\mathbin {\textstyle \coprod _{-_{3}}}-_{1}\colon \mathsf{Fun}\webleft (\mathcal{P},\mathsf{Sets}\webright )\to \mathsf{Sets}_{*}, \]
where $\mathcal{P}$ is the category that looks like this:
In particular, the action on morphisms of $-_{1}\mathbin {\textstyle \coprod _{-_{3}}}-_{1}$ is given by sending a morphism
in $\mathsf{Fun}\webleft (\mathcal{P},\mathsf{Sets}_{*}\webright )$ to the morphism of pointed sets
\[ \xi \colon \webleft (X\mathbin {\textstyle \coprod _{Z}}Y,p_{0}\webright )\overset {\exists !}{\to }\webleft (X'\mathbin {\textstyle \coprod _{Z'}}Y',p'_{0}\webright ) \]
given by
\[ \xi \webleft (p\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \phi \webleft (x\webright ) & \text{if $p=\webleft [\webleft (0,x\webright )\webright ]$},\\ \psi \webleft (y\webright ) & \text{if $p=\webleft [\webleft (1,y\webright )\webright ]$} \end{cases} \]
for each $p\in X\mathbin {\textstyle \coprod _{Z}}Y$, which is the unique morphism of pointed sets making the diagram
commute.