Internalizations of Decorated Bicategories via π ₂-Indexings

We treat the problem of lifting bicategories into double categories through categories of vertical morphisms. We consider structures on decorated 2-categories allowing us to formally implement arguments of sliding certain squares along vertical subdivisions in double categories. We call these structures π ₂ π 2 -indexings. We present a construction associating, to every π ₂ π 2 -indexing on a decorated 2-category, a length 1 double internalization.