Vector bundles with connections over the same manifold $M$ make up a category. Indeed, let $E, E' \twoheadrightarrow M$ be vector bundles with connections $\nabla$ and $\nabla'$. A morphism between vector bundle with connections is a vector bundle morphism $F: E \to E'$ such that for all sections $s \in \Gamma (E)$ it holds $F(\nabla s)=\nabla' F(s)$.
Does there exist a similar category for vector bundles over arbitrary (non-fixed) manifolds? I was hoping that the above construction could be generalized in such a way that the natural morphism on the pull-back vector bundle induces a morphism on the pull-back vector bundle with the pull-back connection, but I am pretty sure the answer is no, as there is no sensible way to pull-back a section through a generic bundle morphism.