I'm wondering whether constructions on vector bundles, such as the Whitney sum or tensor product of two bundles, satisfy some kind of universal property.
What I mean by "some kind of universal property" is kind of vague, but I'm imagining something along the lines of: any bundle constructed from two vector bundles which has the same fibers as the Whitney sum of those bundles is the same as the Whitney sum up to unique isomorphism.
Is there any such universal property that we could formulate and assert for constructions such as the Whitney sum and tensor product of two bundles?