I am trying to make sense of the following statment (from Atiyah's K-theory book)
Suppose $V$ and $W$ 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.