I saw the statement: $E^* \otimes F \cong \text{Hom} (E,F)$ for $E,F$ vector bundles over a manifold $M$, and I want to prove it. My first problem is that I do not know what is the relation $\cong$ here, since the LHS is a vector bundle and the RHS is a vector space. Maybe it just means "bijection"? Despite that, I tried this:
Recall that an element of $\Phi\in \text{Hom}(E,F)$ is a smooth map $\Phi:E\rightarrow F$ such that $p_E=p_F\circ\Phi$.
Now, $$E^*\otimes F = \bigsqcup_{x\in M} E_x^ *\otimes F_x\cong \bigsqcup_{x\in M} \text{Hom} (E_x,F_x)$$
(I do not know how to make precise sense of this $\cong$, just as above...)
And now I want to show that $\bigsqcup_{x\in M} \text{Hom} (E_x,F_x)$ is bijective to $\text{Hom}(E,F)$. But after thinking a bit about it, this just seems false.