Firstly, we note that indeed $\webleft [x_{0}\webright ]=\webleft [y_{0}\webright ]$, as we have
\begin{align*} x_{0} & = f\webleft (z_{0}\webright ),\\ y_{0} & = g\webleft (z_{0}\webright ) \end{align*}
since $f$ and $g$ are morphisms of pointed sets, with the relation $\mathord {\sim }$ on $X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}_{Z}Y$ then identifying $x_{0}=f\webleft (z_{0}\webright )\sim g\webleft (z_{0}\webright )=y_{0}$.
We now claim that $\webleft (X\mathbin {\textstyle \coprod _{Z}}Y,p_{0}\webright )$ is the categorical pushout of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ over $\webleft (Z,z_{0}\webright )$ with respect to $\webleft (f,g\webright )$ in $\mathsf{Sets}_{*}$. First we need to check that the relevant pushout diagram commutes, i.e. that we have
Indeed, given $z\in Z$, we have
\begin{align*} \webleft [\mathrm{inj}_{1}\circ f\webright ]\webleft (z\webright ) & = \mathrm{inj}_{1}\webleft (f\webleft (z\webright )\webright )\\ & = \webleft [\webleft (0,f\webleft (z\webright )\webright )\webright ]\\ & = \webleft [\webleft (1,g\webleft (z\webright )\webright )\webright ]\\ & = \mathrm{inj}_{2}\webleft (g\webleft (z\webright )\webright )\\ & = \webleft [\mathrm{inj}_{2}\circ g\webright ]\webleft (z\webright ),\end{align*}
where $\webleft [\webleft (0,f\webleft (z\webright )\webright )\webright ]=\webleft [\webleft (1,g\webleft (z\webright )\webright )\webright ]$ by the definition of the relation $\mathord {\sim }$ on $X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y$ (the coproduct of unpointed sets of $X$ and $Y$). Next, we prove that $X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}_{Z}Y$ satisfies the universal property of the pushout. Suppose we have a diagram of the form
in $\mathsf{Sets}_{*}$. Then there exists a unique morphism of pointed sets
\[ \phi \colon \webleft (X\mathbin {\textstyle \coprod _{Z}}Y,p_{0}\webright )\to \webleft (P,*\webright ) \]
making the diagram
commute, being uniquely determined by the conditions
\begin{align*} \phi \circ \mathrm{inj}_{1} & = \iota _{1},\\ \phi \circ \mathrm{inj}_{2} & = \iota _{2}\end{align*}
via
\[ \phi \webleft (p\webright )=\begin{cases} \iota _{1}\webleft (x\webright ) & \text{if $x=\webleft [\webleft (0,x\webright )\webright ]$,}\\ \iota _{2}\webleft (y\webright ) & \text{if $x=\webleft [\webleft (1,y\webright )\webright ]$} \end{cases} \]
for each $p\in X\mathbin {\textstyle \coprod _{Z}}Y$, where the well-definedness of $\phi $ is proven in the same way as in the proof of Chapter 2: Constructions With Sets, Definition 2.2.4.1.1. Finally, we show that $\phi $ is indeed a morphism of pointed sets, as we have
\begin{align*} \phi \webleft (p_{0}\webright ) & = \phi \webleft (\webleft [\webleft (0,x_{0}\webright )\webright ]\webright )\\ & = \iota _{1}\webleft (x_{0}\webright )\\ & = *, \end{align*}
or alternatively
\begin{align*} \phi \webleft (p_{0}\webright ) & = \phi \webleft (\webleft [\webleft (1,y_{0}\webright )\webright ]\webright )\\ & = \iota _{2}\webleft (y_{0}\webright )\\ & = *, \end{align*}
where we use that $\iota _{1}$ (resp. $\iota _{2}$) is a morphism of pointed sets.
