Categorical diagonal (${\Delta }_A$)

For a binary product in a category, the diagonal ${\Delta }_A:A\to A\times A$ is the unique map whose two projection composites are $1_A$. It is a split monomorphism, since either projection is a left inverse. The diagonal converts equality of two maps into a lifting problem: $⟨u,v⟩$ factors through ${\Delta }_A$ exactly when $u=v$.