  In this paper we present cartesian structure for symmetric Gray-monoidal
double categories. To do this we first introduce locally cubical Gray
categories, which are three-dimensional categorical structures analogous to
classical, locally globular, Gray categories. The motivating example comprises
double categories themselves, together with their functors, transformations,
and modifications. A one-object locally cubical Gray category is a
Gray-monoidal double category. Braiding, syllepsis, and symmetry for these is
introduced in a manner analogous to that for 2-categories. Adding cartesian
structure requires the introduction of doubly-lax functors of double categories
to manage the order of copies. The resulting theory is algebraically rather
complex, largely due to the bureaucracy of linearizing higher-dimensional
boundary constraints. Fortunately, it has a relatively simple and compelling
representation in the graphical calculus of surface diagrams, which we present.
