Definition Iwss triple category [efr-003Q]

Need a good name here!

An iwss triple category (isofibrant weak-strict-strict) is a bisimplicial category \mathbb {C}: \Delta ^\mathrm {op} \times \Delta ^\mathrm {op} \to \mathsf {Cat} so that for each n, \mathbb {C}[n,-] is a fibrant strict Segal category, and \mathbb {C}[-,n] is a fibrant weak Segal category.

1. A object of the triple category \mathbb {C} is an object of \mathbb {C}[0,0].
2. The morphisms of \mathbb {C}[0,0] are called the vertical morphisms of \mathbb {C}. They form a category denoted \mathbb {C}_{0,0,-}
3. The objects of \mathbb {C}[0,1] are called the horizontal morphisms of \mathbb {C}. They form a category denoted \mathbb {C}_{0,-,0}
4. The objects of \mathbb {C}[1,0] are called the transverse morphisms of \mathbb {C}, or the protomorphisms.
5. The vertical and horizontal morphisms form a strict double category (the one with nerve \mathbb {C}[0,-]), whose 2-cells are the morphisms of \mathbb {C}[0,1]. We call this the planar double category, and denote it \mathbb {C}_{0,-,-}
6. The vertical and transverse morphisms form a pseudo double category (the one with nerve \mathbb {C}[-,0]), whose 2-cells are the morphisms of \mathbb {C}[1,0]. We call this the vertical double category, and denote it \mathbb {C}_{-,0,-}
7. The horizontal transverse morphisms also form a pseudo double category, which has nerve \operatorname {\mathbf {ob}} \mathbb {C}[-,-]---its 2-cells are the objects of \mathbb {C}[1,1]. We call this the horizontal double category, and denote it \mathbb {C}_{-,-,0}
8. For each of the three classes of morphism, there is a double category with it as the objects, the two classes of 2-cells involving it as the 1-cells, and the morphisms of \mathbb {C}[1,1] as the 2-cells. These are denoted \mathbb {C}_{1,-,-}, etc. There is no good way to name them.
9. The globular horizontal 2-cell isomorphisms, and the globular vertical 2-cell isomorphisms, are the same (in the sense that each has a companion in \mathbb {C}_{1,-,-}, and hence the two sets of isomorphisms between any two transverse morphisms are canonically and functorially in bijection). Taking either of these sets as the 2-cells, there is a bicategory with the transverse morphisms as its morphisms, denoted \mathbb {C}_{0,-,0}