Given $E,F$ vector bundles over a manifold $M$, I would like to know a proof of $\Gamma(E\otimes F) = \Gamma(E) \otimes_{C^\infty(M)} \Gamma(F)$. Where $\Gamma$ denotes the smooth sections over $M$. Thanks
This question is similar to      and    except that I am concerned with smooth manifolds. If $M$ is a smooth manifold and $V$ and $W$ are (smooth, real) vector bundles on $M$, are there reasonable conditions that ensure that the map $\Gamma(M,V)\otimes_{\mathscr{O}_M(M)}\Gamma(M,W)\rightarrow\Gamma(M,V\otimes W)$ is an isomorphism? Equivalently, are there any reasonable conditions under which taking global sections of finite locally free $\mathscr{O}_M$-modules commutes with tensor product?  I don't really know much smooth manifold theory, and my intuition is mostly derived from algebraic geometry, where the analogous question, as is indicated in the questions I linked to, is not a simple one, even for invertible sheaves. I'd always assumed the situation for smooth manifolds would be the same, but I was just looking at the Wikipedia page on vector-valued differential forms, and it seems (unless I'm misunderstanding which is certainly possible) to claim there that for $V$ an arbitrary vector bundle and $W$ the $p$-th exterior power of the cotangent bundle, $p\geq 1$, the answer to my question is "yes," with the justification being that "$\Gamma(M,-)$ is a monoidal functor." Looking at the definition of a monoidal functor, it seems to me that all this implies is that there is a map from the tensor product of the global sections to the global sections of the tensor product (which is clear from the definitions) and not that this map is an isomorphism.
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