I'm reading this notes, and at the page 16 he defines a flat structure on a vector bundle as follows:
A flat structure on a vector bundle $E$ is a family $\{\psi_U \}_{U∈\mathcal{U}}$ of trivializations, where $\mathcal{U}$ is an open cover of $M$, such that the changes of trivializations $g_{U,V}$ are locally constants.
My question came when he asked:
If ($E$, $\{\psi_U \}_{U∈\mathcal{U}}$) and ($E'$, $\{\psi_U' \}_{U'∈\mathcal{U}'}$) are two flat bundles, what is a morphism between $E$ and $E'$? an isomorphism?
For me the "natural notion" of morphism of vector bundles is the canonical one, that given ($E_1$, $\pi_1$, $M$) and ($E_2$, $\pi_2$, $M$) two fiber bundles a morphism between them is a function $\phi:E_1\rightarrow E_2$ such that $\pi_2\circ \phi=\pi_1$ and its restriction to each fiber is a linear map.
Why in this case do we use a different notion of morphism? And what is this notion?