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