Projection map

A projection map from a Cartesian product selects specified coordinates, for example $p_j:{\prod }_i{X}_i\to X_j$ with $p_j((x_i{)}_i)=x_j$.

For a coordinate projection map onto $X_i$, write $X_j$ for the other factor. The map is surjective exactly when $X_i$ is empty or $X_j$ is nonempty. It is injective exactly when $X_i$ is empty or $X_j$ has at most one element. In particular, a surjection onto an empty Cartesian product need not induce a surjection onto a nonempty factor.