Functoriality. The assignment $\webleft (A,B,C,f,g\webright )\mapsto A\mathbin {\textstyle \coprod _{f,C,g}}B$ 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 map $\xi \colon A\mathbin {\textstyle \coprod _{C}}B\overset {\exists !}{\to }A'\mathbin {\textstyle \coprod _{C'}}B'$ given by
\[ \xi \webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \phi \webleft (a\webright ) & \text{if $x=\webleft [\webleft (0,a\webright )\webright ]$},\\ \psi \webleft (b\webright ) & \text{if $x=\webleft [\webleft (1,b\webright )\webright ]$} \end{cases} \]
for each $x\in A\mathbin {\textstyle \coprod _{C}}B$, which is the unique map making the diagram
commute.