What are the projection maps? I've read that projection maps are an important type of maps whose domain is a product of n possibly different sets. My question is that why do they name them "projection" maps? What are we projecting exactly? Yes we are mapping or relating the product of the sets to an output (thats what a mapping already is ) but what are we projecting?

Generally, a projection map "projects" elements onto a lower dimensional subspace which is a product of some subset of those sets. For example, consider the map $(x,y) \mapsto x$ from $\mathbb{R}\times\mathbb{R} \to \mathbb{R}$, which projects onto the first coordinate. If you were to express this graphically, you would see that you can take any point and project it vertically onto its $x$-coordinate, almost as if it were the "shadow" of the point along this vertoca; line.
