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    $\begingroup$ Having said that, while I am not the downvoter and find your post interesting, I don't really think it addresses OP's question. My impression is that the OP understands the definition of vector bundles and their tensor product and is asking specifically for a geometric interpretation of $E\otimes F$ akin to how $E \oplus F$ is the ground for topological K-theory and akin to the Picard group in the case of line bundles. $\endgroup$ Commented Jun 26, 2023 at 12:42
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    $\begingroup$ I agree that the standard set-theoretic definition of a function isn't quite a perfect fit for all use cases, in particular the need to specify a single codomain to contain all outputs is slightly unnatural sometimes (even if the axiom schema of replacement guarantees that it can always be done). As for your other point, the OP already appears to have some existing geometric intuition for tensor products of vector spaces; I hold that one just needs to relativize this intuition (replacing vector spaces/points with vector bundles/sections etc.) over $X$ to answer the question. $\endgroup$ Commented Jun 26, 2023 at 16:32
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    $\begingroup$ This does not really answer the question. I am sure that OP understands the intuition of doing operations on a vector bundle pointwise. However, the actual construction is much more involved than that: one needs to provide local trivialisations which respect the differential geometry of the manifold of the base points. If one just provides the set of all fibres of a vector bundle, which basically is $M \times V$, one has not done any geometry at all! To turn $M \times V$ into the correct manifold, one has to get into the nitty gritty of sets, I fear. $\endgroup$ Commented Jun 26, 2023 at 16:56
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    $\begingroup$ As mentioned in Paul's answer, one can encode the geometry entirely through through the sheaf of sections, which is manipulated via the pointwise operations; local trivializations are not really required other than for setting up foundations. For instance the sections of a tensor product $E_1 \otimes E_2$ of bundles are locally the linear combinations of formal products of sections of $E_1$ and $E_2$ separately, modulo bilinearity relations; this is just the pointwise application of the usual tensor product, but relativized so that points in a vector space become sections of a vector bundle. $\endgroup$ Commented Jun 26, 2023 at 19:27
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    $\begingroup$ If you want smooth geometry, you require the sections to be smooth. If you want complex geometry, you require the sections to be holomorphic. If you want discrete geometry, you allow arbitrary sections. And so forth. One works in whatever topos is suited best for one's application. But regardless of the choice of topos, the operations are pointwise, and are simply the relativizations of the absolute version of the operation to the topos at hand. $\endgroup$ Commented Jun 26, 2023 at 19:31