Functoriality. The assignment $\webleft (A,B,C,f,g\webright )\mapsto A\times _{f,C,g}B$ defines a functor
\[ -_{1}\times _{-_{3}}-_{1}\colon \mathsf{Fun}\webleft (\mathcal{P},\mathsf{Sets}\webright )\to \mathsf{Sets}, \]
where $\mathcal{P}$ is the category that looks like this:
In particular, the action on morphisms of $-_{1}\times _{-_{3}}-_{1}$ is given by sending a morphism
in $\mathsf{Fun}\webleft (\mathcal{P},\mathsf{Sets}\webright )$ to the map $\xi \colon A\times _{C}B\overset {\exists !}{\to }A'\times _{C'}B'$ given by
\[ \xi \webleft (a,b\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\phi \webleft (a\webright ),\psi \webleft (b\webright )\webright ) \]
for each $\webleft (a,b\webright )\in A\times _{C}B$, which is the unique map making the diagram
commute.