Let $\webleft\{ \webleft (X_{i},x^{i}_{0}\webright )\webright\} _{i\in I}$ be a family of pointed sets.
The product of $\smash {\webleft\{ \webleft (X_{i},x^{i}_{0}\webright )\webright\} _{i\in I}}$ is the pair $\smash {\webleft (\webleft (\prod _{i\in I}X_{i},\webleft (x^{i}_{0}\webright )_{i\in I}\webright ),\webleft\{ \text{pr}_{i}\webright\} _{i\in I}\webright )}$ consisting of:
- The Limit. The pointed set $\webleft (\prod _{i\in I}X_{i},\webleft (x^{i}_{0}\webright )_{i\in I}\webright )$.
- The Cone. The collection
\[ \webleft\{ \text{pr}_{i} \colon \webleft (\prod _{i\in I}X_{i},\webleft (x^{i}_{0}\webright )_{i\in I}\webright )\to \webleft (X_{i},x^{i}_{0}\webright )\webright\} _{i\in I} \]
of maps given by
\[ \text{pr}_{i}\webleft (\webleft (x_{j}\webright )_{j\in I}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x_{i} \]
for each $\webleft (x_{j}\webright )_{j\in I}\in \prod _{i\in I}X_{i}$ and each $i\in I$.
We claim that $\smash {\webleft (\prod _{i\in I}X_{i},\webleft (x^{i}_{0}\webright )_{i\in I}\webright )}$ is the categorical product of $\smash {\webleft\{ \webleft (X_{i},x^{i}_{0}\webright )\webright\} _{i\in I}}$ in $\mathsf{Sets}_{*}$. Indeed, suppose we have, for each $i\in I$, a diagram of the form
in $\mathsf{Sets}_{*}$. Then there exists a unique morphism of pointed sets
\[ \phi \colon \webleft (P,*\webright )\to \webleft (\prod _{i\in I}X_{i},\webleft (x^{i}_{0}\webright )_{i\in I}\webright ) \]
making the diagram
commute, being uniquely determined by the condition $\text{pr}_{i}\circ \phi =p_{i}$ for each $i\in I$ via
\[ \phi \webleft (x\webright )=\webleft (p_{i}\webleft (x\webright )\webright )_{i\in I} \]
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_{i}\webleft (*\webright )\webright )_{i\in I}\\ & = \webleft (x^{i}_{0}\webright )_{i\in I},\end{align*}
where we have used that $p_{i}$ is a morphism of pointed sets for each $i\in I$.
