The Clowder Project

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

1. 1. We think of $\prod _{i\in I}{A}_i$ as the set whose elements are $I$-indexed collections $\left(a_i\right){}_{i\in I}$ with $a_i\in A_i$ for each $i\in I$.
2. 2. We view the projection maps
   $$\{{\text{pr}}_i:\prod _{i\in I}{A}_i\to A_i\}{}_{i\in I}$$

   as being given by

   $${\text{pr}}_i\left(\left(a_j\right){}_{j\in I}\right)\stackrel{\text{def}}{=}{a}_i$$

   for each $\left(a_j\right){}_{j\in I}\in \prod _{i\in I}{A}_i$ and each $i\in I$.