Associated bundles in restriction category world

In classical differential geometry, principal bundles are locally the cartesian product of a manifold and a group, vector bundles are locally the cartesian product of a manifold and a vector space. It is a classical result that there is a one to one correspondance between the category of principal ${\mathrm{GL}}_n$-bundles and the category of vector bundles with fiber ${ℝ}^n$.

In [1] Cockett and Cruttwell generalized vector bundles as differential bundles in tangent categories. In [2], Cockett and I generalized principal bundles to restriction categories and in particular to tangent restriction categories.

Does the correspondance between principal and vector bundles hold up in this generalized setup?

It does not. I will give an example of a restriction category in which there are more differential bundles than principal bundles. However one direction of the correspondance still holds. The construction that sends a principal bundle to a vector bundle is known as the associated bundle. We will perform this construction in the setting of tangent restriction categories, obtain a functor and describe its properties. This does not just recover a classical result from differential geometry, it also provides a strategy for constructing examples of differential bundles in tangent restriction categories.