The product of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ is the pair consisting of:
- The Limit. The pointed set $\webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )$.
- The Cone. The morphisms of pointed sets
\begin{align*} \text{pr}_{1} & \colon \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )\to \webleft (X,x_{0}\webright ),\\ \text{pr}_{2} & \colon \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )\to \webleft (Y,y_{0}\webright ) \end{align*}
defined by
\begin{align*} \text{pr}_{1}\webleft (x,y\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x,\\ \text{pr}_{2}\webleft (x,y\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}y \end{align*}
for each $\webleft (x,y\webright )\in X\times Y$.
We claim that $\webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )$ is the categorical product of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ in $\mathsf{Sets}_{*}$. Indeed, suppose we have a diagram of the form
in $\mathsf{Sets}_{*}$. Then there exists a unique morphism of pointed sets
\[ \phi \colon \webleft (P,*\webright )\to \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright ) \]
making the diagram
commute, being uniquely determined by the conditions
\begin{align*} \text{pr}_{1}\circ \phi & = p_{1},\\ \text{pr}_{2}\circ \phi & = p_{2} \end{align*}
via
\[ \phi \webleft (x\webright )=\webleft (p_{1}\webleft (x\webright ),p_{2}\webleft (x\webright )\webright ) \]
for each $x\in P$. Note that this is indeed a morphism of pointed sets, as we have
\begin{align*} \phi \webleft (*\webright ) & = \webleft (p_{1}\webleft (*\webright ),p_{2}\webleft (*\webright )\webright )\\ & = \webleft (x_{0},y_{0}\webright ),\end{align*}
where we have used that $p_{1}$ and $p_{2}$ are morphisms of pointed sets.
