Empty-set cases for Cartesian projections ($p_i:X_1\times X_2\to X_i$)

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.