Is $\Gamma(E\boxtimes F) \cong \Gamma(\pi_M^*E)\otimes_{C^\infty(M\times M)}\Gamma(\pi_N^*F)$?

Question

Let $E\longrightarrow M$ and $F\longrightarrow N$ be two smooth vector bundles. Is it true that $$\Gamma(E\boxtimes F) \cong \Gamma(\pi_M^*E)\otimes_{C^\infty(M\times M)}\Gamma(\pi_N^*F)?$$ $\boxtimes$ denotes external tensor product of vector bundles. $M\times N \xrightarrow{\pi_M}M$ and $M\times N \xrightarrow{\pi_N}N$ are canonical projections. $\Gamma$ is the smooth section functor from vector bundles to modules over smooth scalar functions. I haven't found any literature covering external tensor product.