The product1 of $\left\{ A_{i}\right\} _{i\in I}$ is the product of $\left\{ A_{i}\right\} _{i\in I}$ in $\mathsf{Sets}$ as in 
, .
- 1Further Terminology: Also called the Cartesian product of $\left\{ A_{i}\right\} _{i\in I}$.
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The Limit. The set $\prod _{i\in I}A_{i}$ defined by
\[ \prod _{i\in I}A_{i} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ f\in \mathsf{Sets}\left(I,\bigcup _{i\in I}A_{i}\right)\ \middle |\ \begin{aligned} & \text{for each $i\in I$, we}\\ & \text{have $f(i)\in A_{i}$}\end{aligned} \right\} . \] -
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
\[ \operatorname {\mathrm{\mathrm{pr}}}_{i}(f)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f(i) \]for each $f\in \prod _{i\in I}A_{i}$ and each $i\in I$.
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We think of $\prod _{i\in I}A_{i}$ as the set whose elements are $I$-indexed collections $(a_{i})_{i\in I}$ with $a_{i}\in A_{i}$ for each $i\in I$.
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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
\[ \operatorname {\mathrm{\mathrm{pr}}}_{i}((a_{j})_{j\in I})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}a_{i} \]for each $(a_{j})_{j\in I}\in \prod _{i\in I}A_{i}$ and each $i\in I$.
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Functoriality. The assignment $\left\{ A_{i}\right\} _{i\in I}\mapsto \prod _{i\in I}A_{i}$ defines a functor
\[ \prod _{i\in I}\colon \mathsf{Fun}(I_{\mathsf{disc}},\mathsf{Sets})\to \mathsf{Sets} \]where
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Action on Objects. For each $(A_{i})_{i\in I}\in \operatorname {\mathrm{Obj}}(\mathsf{Fun}(I_{\mathsf{disc}},\mathsf{Sets}))$, we have
\[ \left[\prod _{i\in I}\right]((A_{i})_{i\in I})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\prod _{i\in I}A_{i} \] -
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Action on Morphisms. For each $(A_{i})_{i\in I},(B_{i})_{i\in I}\in \operatorname {\mathrm{Obj}}(\mathsf{Fun}(I_{\mathsf{disc}},\mathsf{Sets}))$, the action on $\operatorname {\mathrm{Hom}}$-sets
\[ \left(\prod _{i\in I}\right)_{(A_{i})_{i\in I},(B_{i})_{i\in I}} \colon \operatorname {\mathrm{Nat}}((A_{i})_{i\in I},(B_{i})_{i\in I})\to \mathsf{Sets}\left(\prod _{i\in I}A_{i},\prod _{i\in I}B_{i}\right) \]of $\prod _{i\in I}$ at $((A_{i})_{i\in I},(B_{i})_{i\in I})$ is defined by sending a map
\[ \left\{ f_{i}\colon A_{i}\to B_{i} \right\} _{i\in I} \]in $\operatorname {\mathrm{Nat}}((A_{i})_{i\in I},(B_{i})_{i\in I})$ 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
\[ \left[\prod _{i\in I}f_{i}\right]((a_{i})_{i\in I}) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(f_{i}(a_{i}))_{i\in I} \]for each $(a_{i})_{i\in I}\in \prod _{i\in I}A_{i}$.
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4.1.2 Products of Families of Sets
Let $\left\{ A_{i}\right\} _{i\in I}$ be a family of sets.
Concretely, the product of $\left\{ A_{i}\right\} _{i\in I}$ is the pair $(\prod _{i\in I}A_{i},\left\{ \operatorname {\mathrm{\mathrm{pr}}}_{i}\right\} _{i\in I})$ consisting of:
Proof of Construction 4.1.2.1.2.
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
\[ \phi (x)=(p_{i}(x))_{i\in I} \]
for each $x\in P$.
Less formally, we may think of Cartesian products and projection maps as follows:
Let $\left\{ A_{i}\right\} _{i\in I}$ be a family of sets.
Proof of Proposition 4.1.2.1.4.
Item 1: Functoriality
This follows from , of .
