# Free Globularly Generated Double Categories II: The Canonical Double Projection

## TL;DR

This paper introduces the canonical double projection in free globularly generated double categories, enabling translation of data and extending key operations, thus advancing the understanding of their structure and applications.

## Contribution

It defines the canonical double projection and demonstrates its use in extending the Haagerup standard form and Connes fusion, also establishing a left adjoint relationship with decorated horizontalization.

## Key findings

- Canonical double projection effectively translates data between categories.
- Extended the Haagerup standard form and Connes fusion to possibly-infinite index morphisms.
- Proved the free globularly generated double category construction is left adjoint to decorated horizontalization.

## Abstract

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.