  This is the second installment of a two part series of papers studying free
globularly generated double categories. We introduce the canonical double
projection construction. The canonical double projection translates information
from free globularly generated double categories to double categories defined
through the same set of globular and vertical data. We use the canonical double
projection to define compatible formal linear functorial extensions of the
Haagerup standard form and the Connes fusion operation to possibly-infinite
index morphisms between factors. We use the canonical double projection to
prove that the free globularly generated double category construction is left
adjoint to decorated horizontalization. We thus interpret free globularly
generated double categories as formal decorated analogs of double categories of
quintets and as generators for internalizations.
