I have a (perhaps stupid) question dealing with vector bundles.
In most textbooks, a morphism of real vector bundles $f:\xi\to \eta$ over a fixed base space $X$ is said to be a $\textsf{monomorphism}$ if the total space map $f:E\to E^{\prime}$ is (fiberwise) injective. (Epimorphisms are analogously defined fiberwisely.)
What I want to check is that this definition matches with the definition of $\textsf{monomorphism}$s in the category of vector bundles over $X$ (let us denote this as $\textbf{VB}_{X}$), namely a morphism which can be cancelled from left. Vector bundle morphisms with injective total space maps are obviously monomorphisms in this category-theoretic sense. However, I can't show the converse, that $\textbf{VB}_{X}$-monomorphisms necessarily have injective total space maps (I can't find a thing like an "indicator sheaf" for sheaves in this vector bundle case...). I now even suspect that these two notions are not equivalent, but I can't find a good counterexample. Any helps would be appreciated!