Categorical realization of labeled Poincaré–Reeb V-digraphs

Formulate a category that generalizes or respects the labeled Poincaré–Reeb V-digraphs introduced in Theorem 1 and Section 3, and realize every object in that category by explicitly obtaining an NI arrangement of circles $(\mathcal{S},D_{\mathcal{S}})$ up to isomorphism in the category.

Background

The paper introduces labels on the vertices and edges of Poincaré–Reeb V-digraphs associated with normally inductive arrangements of circles. These labels record information about the circles and their $(j,j+1)\frac{\pi}{4}$-arcs that occur in the corresponding fibers of the coordinate projections.

The authors ask for an abstract categorical framework capturing these labeled graph structures and for a realization theorem showing that every object of the resulting category arises from an explicitly constructed NI arrangement of circles, modulo categorical isomorphism.