Does anybody know a reference for fact that the Functor $\Gamma^\infty$, assigning to every smooth vector bundle $\mathcal{E}\to M$ the $C^\infty(M)$-module $\Gamma^\infty(\mathcal{E})$ of smooth sections in $\mathcal{E}$, is exact?
Edit: to be more precise, we are considering the category of smooth vector bundles of finite rank over a fixed, smooth, finite-dimensional and second-countable base manifold $M$. A morphism $F:\mathcal{E}\to\mathcal{F}$ is a smooth, fiberwise linear map $F$ between the total spaces of $\mathcal{E}$ and $\mathcal{F}$, such that $\pi^\mathcal{F}\circ F=\pi^\mathcal{E}$. Applying $\Gamma^\infty$ then gives the $C^\infty$-linear map $$\Gamma^\infty F:\Gamma^\infty(\mathcal{E})\ni s\mapsto F\circ s\in\Gamma^\infty(\mathcal{F}).$$