  We classify surjective lattice homomorphisms $W\to W'$ between the weak
orders on finite Coxeter groups. Equivalently, we classify lattice congruences
$\Theta$ on $W$ such that the quotient $W/\Theta$ is isomorphic to $W'$.
Surprisingly, surjective homomorphisms exist quite generally: They exist if and
only if the diagram of $W'$ is obtained from the diagram of $W$ by deleting
vertices, deleting edges, and/or decreasing edge labels. A surjective
homomorphism $W\to W'$ is determined by its restrictions to rank-two standard
parabolic subgroups of $W$. Despite seeming natural in the setting of Coxeter
groups, this determination in rank two is nontrivial. Indeed, from the
combinatorial lattice theory point of view, all of these classification results
should appear unlikely a priori. As an application of the classification of
surjective homomorphisms between weak orders, we also obtain a classification
of surjective homomorphisms between Cambrian lattices and a general
construction of refinement relations between Cambrian fans.
