Internalizations of decorated bicategories via $\pi_2$-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 $\pi_2$-indexings. We present a construction associating, to every $\pi_2$-indexing on a decorated 2-category, a length 1 double internalization.