The product of $\smash {\left\{ \webleft (X_{i},x^{i}_{0}\webright )\right\} _{i\in I}}$ is the product of $\left\{ \webleft (X_{i},x^{i}_{0}\webright )\right\} _{i\in I}$ in $\mathsf{Sets}_{*}$ as in 
, .
6.2.2 Products of Families of Pointed Sets
Let $\left\{ \webleft (X_{i},x^{i}_{0}\webright )\right\} _{i\in I}$ be a family of pointed sets.
Concretely, the product of $\smash {\left\{ \webleft (X_{i},x^{i}_{0}\webright )\right\} _{i\in I}}$ is the pair $\smash {\webleft (\webleft (\prod _{i\in I}X_{i},\webleft (x^{i}_{0}\webright )_{i\in I}\webright ),\left\{ \operatorname {\mathrm{\mathrm{pr}}}_{i}\right\} _{i\in I}\webright )}$ consisting of:
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The Limit. The pointed set $\webleft (\prod _{i\in I}X_{i},\webleft (x^{i}_{0}\webright )_{i\in I}\webright )$.
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The Cone. The collection
\[ \left\{ \operatorname {\mathrm{\mathrm{pr}}}_{i} \colon \webleft (\prod _{i\in I}X_{i},\webleft (x^{i}_{0}\webright )_{i\in I}\webright )\to \webleft (X_{i},x^{i}_{0}\webright )\right\} _{i\in I} \]of maps given by
\[ \operatorname {\mathrm{\mathrm{pr}}}_{i}\webleft (\webleft (x_{j}\webright )_{j\in I}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x_{i} \]for each $\webleft (x_{j}\webright )_{j\in I}\in \prod _{i\in I}X_{i}$ and each $i\in I$.
Proof of Construction 6.2.2.1.2.
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 $\smash {\left\{ \webleft (X_{i},x^{i}_{0}\webright )\right\} _{i\in I}}$ in $\mathsf{Sets}_{*}$. Indeed, suppose we have, for each $i\in I$, a diagram of the form
\[ \phi \colon \webleft (P,*\webright )\to \webleft (\prod _{i\in I}X_{i},\webleft (x^{i}_{0}\webright )_{i\in I}\webright ) \]
making the diagram
\[ \phi \webleft (x\webright )=\webleft (p_{i}\webleft (x\webright )\webright )_{i\in I} \]
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 $i\in I$.
Let $\left\{ \webleft (X_{i},x^{i}_{0}\webright )\right\} _{i\in I}$ be a family of pointed sets.
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Functoriality. The assignment $\left\{ \webleft (X_{i},x^{i}_{0}\webright )\right\} _{i\in I}\mapsto \webleft (\prod _{i\in I}X_{i},\webleft (x^{i}_{0}\webright )_{i\in I}\webright )$ defines a functor
\[ \prod _{i\in I}\colon \mathsf{Fun}\webleft (I_{\mathsf{disc}},\mathsf{Sets}_{*}\webright )\to \mathsf{Sets}_{*}. \]
Proof of Proposition 6.2.2.1.3.
Item 1: Functoriality
This follows from , of .
