  The box product and its associated box exponential are characterized for the
categories of quivers (directed graphs), multigraphs, set system hypergraphs,
and incidence hypergraphs. It is shown that only the quiver case of the box
exponential can be characterized via homs entirely within their own category.
An asymmetry in the incidence hypergraphic box product is rectified via an
incidence dual-closed generalization that effectively treats vertices and edges
as real and imaginary parts of a complex number, respectively. This new
hypergraphic box product is shown to have a natural interpretation as the
canonical box product for graphs via the bipartite representation functor, and
its associated box exponential is represented as homs entirely in the category
of incidence hypergraphs; with incidences determined by incidence-prism
mapping. The evaluation of the box exponential at paths is shown to correspond
to the entries in half-powers of the oriented hypergraphic signless Laplacian
matrix.
