The following could be rather silly question but I haven'd found it stated explicitly; from the other side, it seems to me, that this fact is used often without comments.
The problem is the following: consider vector bundle $S \to M$ and construct the endomorphism vector bundle $End(S) \to M$ where
$$(End(S))_x:=End(S_x).$$
Now take $\Gamma(M,End(S))$ to be the module (over $C(M)$) of continuous sections. Is is true that this module coincides with $End_{C(M)}(\Gamma(M,S))$?