How is a section of $Hom(E,F)$ an example of a differential operator? I'm taking my first course in elliptic operators, and in our first homework assignment we've been given a few examples of basic differential operators $D$ and are asked to calculate their order $k$ and their $k$-symbol. This is the first example:

$D$ is a section of the Hom bundle $Hom(E,F)$, where $E,F\to M$ are arbitrary vector bundles.

I may be missing something obvious, but I don't actually see why this $D$ is an example of a differential operator. Shouldn't we be able to write $D$ as a map $\Gamma(E')\to\Gamma(F')$ for some vector bundles $E',F'\to M$? For instance, another example is a vector field giving an operator $C^{\infty}(M)\to C^{\infty}(M)$, but I immediately recognize that this is the same as a map of global sections $\Gamma(M\times\Bbb R)\to\Gamma(M\times\Bbb R)$.

A section $D$ of $\operatorname{Hom}(E, F)$ defines a linear map $\Gamma(E) \to \Gamma(F)$ given by $s \mapsto D\circ s$.
