Abstract
Consider the intrinsic fundamental group \textit{à la} Grothendieck of a linear category, introduced in \cite{CRS} and \cite{CRS2} using connected gradings. In this article we prove that a full convex subcategory is incompressible, in the sense that the group map between the corresponding fundamental groups is injective. The proof makes essential use of the functoriality of the intrinsic fundamental group, and it is based on the study of the restriction of connected gradings to full subcategories. Moreover, we study in detail the fibre product of coverings and of Galois coverings.