Definition 4.1.2.1.1 (000P): The Product of a Family of Sets

¹Further Terminology: Also called the Cartesian product of $\left\{ A_{i}\right\} _{i\in I}$.

Concretely, the product of $\left\{ A_{i}\right\} _{i\in I}$ is the pair $\webleft (\prod _{i\in I}A_{i},\left\{ \operatorname {\mathrm{\mathrm{pr}}}_{i}\right\} _{i\in I}\webright )$ consisting of:

1.
The Limit. The set $\prod _{i\in I}A_{i}$ defined by

\[ \prod _{i\in I}A_{i} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ f\in \mathsf{Sets}\left(I,\bigcup _{i\in I}A_{i}\right)\ \middle |\ \begin{aligned} & \text{for each $i\in I$, we}\\ & \text{have $f\webleft (i\webright )\in A_{i}$}\end{aligned} \right\} . \]
2.
The Cone. The collection

\[ \left\{ \operatorname {\mathrm{\mathrm{pr}}}_{i} \colon \prod _{i\in I}A_{i}\to A_{i}\right\} _{i\in I} \]

of maps given by

\[ \operatorname {\mathrm{\mathrm{pr}}}_{i}\webleft (f\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (i\webright ) \]

for each $f\in \prod _{i\in I}A_{i}$ and each $i\in I$.

We claim that $\prod _{i\in I}A_{i}$ is the categorical product of $\left\{ A_{i}\right\} _{i\in I}$ 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 map $\phi \colon P\to \prod _{i\in I}A_{i}$ making the diagram

commute, being uniquely determined by the condition $\operatorname {\mathrm{\mathrm{pr}}}_{i}\circ \phi =p_{i}$ for each $i\in I$ via

\[ \phi \webleft (x\webright )=\webleft (p_{i}\webleft (x\webright )\webright )_{i\in I} \]

for each $x\in P$.

Less formally, we may think of Cartesian products and projection maps as follows:

1.
We think of $\prod _{i\in I}A_{i}$ as the set whose elements are $I$-indexed collections $\webleft (a_{i}\webright )_{i\in I}$ with $a_{i}\in A_{i}$ for each $i\in I$.
2.
We view the projection maps

\[ \left\{ \operatorname {\mathrm{\mathrm{pr}}}_{i} \colon \prod _{i\in I}A_{i}\to A_{i}\right\} _{i\in I} \]

as being given by

\[ \operatorname {\mathrm{\mathrm{pr}}}_{i}\webleft (\webleft (a_{j}\webright )_{j\in I}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}a_{i} \]

for each $\webleft (a_{j}\webright )_{j\in I}\in \prod _{i\in I}A_{i}$ and each $i\in I$.

Let $\left\{ A_{i}\right\} _{i\in I}$ be a family of sets.

1.
Functoriality. The assignment $\left\{ A_{i}\right\} _{i\in I}\mapsto \prod _{i\in I}A_{i}$ defines a functor

\[ \prod _{i\in I}\colon \mathsf{Fun}\webleft (I_{\mathsf{disc}},\mathsf{Sets}\webright )\to \mathsf{Sets} \]

where
- •
  Action on Objects. For each $\webleft (A_{i}\webright )_{i\in I}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Fun}\webleft (I_{\mathsf{disc}},\mathsf{Sets}\webright )\webright )$, we have
  
  \[ \left[\prod _{i\in I}\right]\webleft (\webleft (A_{i}\webright )_{i\in I}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\prod _{i\in I}A_{i} \]
- •
  Action on Morphisms. For each $\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Fun}\webleft (I_{\mathsf{disc}},\mathsf{Sets}\webright )\webright )$, the action on $\operatorname {\mathrm{Hom}}$-sets
  
  \[ \left(\prod _{i\in I}\right)_{\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}} \colon \operatorname {\mathrm{Nat}}\webleft (\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\webright )\to \mathsf{Sets}\left(\prod _{i\in I}A_{i},\prod _{i\in I}B_{i}\right) \]
  
  of $\prod _{i\in I}$ at $\webleft (\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\webright )$ is defined by sending a map
  
  \[ \left\{ f_{i}\colon A_{i}\to B_{i} \right\} _{i\in I} \]
  
  in $\operatorname {\mathrm{Nat}}\webleft (\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\webright )$ to the map of sets
  
  \[ \prod _{i\in I}f_{i}\colon \prod _{i\in I}A_{i}\to \prod _{i\in I}B_{i} \]
  
  defined by
  
  \[ \left[\prod _{i\in I}f_{i}\right]\webleft (\webleft (a_{i}\webright )_{i\in I}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f_{i}\webleft (a_{i}\webright )\webright )_{i\in I} \]
  
  for each $\webleft (a_{i}\webright )_{i\in I}\in \prod _{i\in I}A_{i}$.

Item 1: Functoriality

This follows from

The Clowder Project

4.1.2 Products of Families of Sets