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I am trying to make sense of the following statment (from Atiyah's K-theory book)

Suppose $V,W$ are vector spaces, and that $E=X \times V, F=X \times W$ are the corresponding product bundles. Then any homomorphism $\phi:E \to F$ determines a map $\Phi:X \to text{Hom}(V,W)$ by the formula $\phi(x,v) = (x,\Phi(x)v).$

I guess I don't really understand what this is trying to say. Is it that a homomorphism between product bundles over a common base space, induces a map between the associated vector spaces? I guess I am used to seeing a map given explicitly rather than implicitly.

(Tangentially Related)

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In general, if $E$ and $F$ are arbitrary vector bundles over $X$, then by restricting to the fibers over $x \in X$, any vector bundle map $\phi: E \to F$ gives a linear map $\phi_x:E_x \to F_x$. In the case of a product bundle, all fibers of $E$ are $V$ and all fibers of $F$ are $W$. So every $x \in X$ gives us a map $\phi_x: V \to W$. Conversely, given a map $\Phi: X \to Hom(V,W)$ we get a map of vector bundles by $(x,v) \mapsto (x, \Phi(x)(v))$.

Is it that a homomorphism between product bundles over a common base space, induces a map between the associated vector spaces?

A map of product bundles induces many maps between the associated vector spaces; you get one for each $x \in X$ and these maps vary continuously (or smoothly) depending on what type of maps of vector bundles you want to consider. Indeed, maps of product bundles are nothing but maps between the vector spaces parametrized by the base space.

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