Less formally, we may think of Cartesian products and projection maps as follows:
- 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 $i\in I$.
-
We view the projection maps
\[ \webleft\{ \text{pr}_{i} \colon \prod _{i\in I}A_{i}\to A_{i}\webright\} _{i\in I} \]
as being given by
\[ \text{pr}_{i}\webleft (\webleft (a_{j}\webright )_{j\in I}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}a_{i} \]for each $\webleft (a_{j}\webright )_{j\in I}\in \prod _{i\in I}A_{i}$ and each $i\in I$.
