Full and convex linear subcategories are incompressible

Consider the intrinsic fundamental group `a la Grothendieck of a linear category using connected gradings. In this article we prove that any full convex subcategory is incompressible, in the sense that the group map between the corresponding fundamental groups is injective. We start by proving the functoriality of the intrinsic fundamental group with respect to full subcategories, based on the study of the restriction of connected gradings.