  Boij-S\"oderberg theory is the study of two cones: the cone of cohomology
tables of coherent sheaves over projective space and the cone of standard
graded minimal free resolutions over a polynomial ring. Each cone has a
simplicial fan structure induced by a partial order on its extremal rays. We
provide a new interpretation of these partial orders in terms of the existence
of nonzero homomorphisms, for both the general and the equivariant
constructions. These results provide new insights into the families of sheaves
and modules at the heart of Boij-S\"oderberg theory: supernatural sheaves and
Cohen-Macaulay modules with pure resolutions. In addition, our results strongly
suggest the naturality of these partial orders, and they provide tools for
extending Boij-S\"oderberg theory to other graded rings and projective
varieties.
