Category of vector bundles with connections

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.

Let's try to define the category $$\mathcal C$$ of contravariant vector bundles equipped with a connexion, I haven't checked the details and I don't know if it is the right way of seeing it, so be careful. An object in $$\mathcal C$$ is a triple $$(M,E,\nabla)$$ with $$M$$ a manifold, $$E$$ a vector bundle on it and $$\nabla$$ a connexion on $$E$$, that is, a bundle morphism $$E\to E\otimes T^*M$$ satisfying the Leibniz rule. If $$M'$$ is another manifold and $$f\colon M'\to M$$ is a smooth map, there is a functorial morphism $$f^*T^*M\to T^*M'$$, the pullback of forms. Call $$f^*\nabla$$ the composition $$f^*\nabla\colon f^*E\to f^*(E\otimes T^*M)\cong f^*E\otimes f^*T^*M\to f^*E \otimes T^*M'$$ it is a connection on $$f^*E$$.
Now, if $$(M',E',\nabla')$$ is another object in $$\mathcal C$$, a morphism $$(M',E',\nabla')\to (M,E,\nabla)$$ consists of a smooth map $$f\colon M'\to M$$ and a vector bundle morphism $$F: f^*E\to E'$$ such that
commutes. I guess you can also define the category of covariant vector bundles equipped with a connexion by taking the bundle morphism in the opposite direction, $$F: E'\to f^*E$$.