  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.
