Functoriality. The assignment $\webleft (X,Y,Z,f,g\webright )\mapsto X\times _{f,Z,g}Y$ 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 morphism of pointed sets
\[ \xi \colon \webleft (X\times _{Z}Y,\webleft (x_{0},y_{0}\webright )\webright )\overset {\exists !}{\to }\webleft (X'\times _{Z'}Y',\webleft (x'_{0},y'_{0}\webright )\webright ) \]
given by
\[ \xi \webleft (x,y\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\phi \webleft (x\webright ),\psi \webleft (y\webright )\webright ) \]
for each $\webleft (x,y\webright )\in X\times _{Z}Y$, which is the unique morphism of pointed sets making the diagram
commute.