In the answer of this post Fundamental result on the projective tensor product of sections of a vector bundle we have $\Gamma(M, V\times M) \cong C^\infty(M) \otimes V$
Where $\Gamma(M, V\times M)$ is the space of section of a trivial vector bundle $E$ with base space $M$ and fiber $V$
How to proof this result ? Is there any reference that proof it?
If $f \in C^\infty(M)$ and $v \in V$, then $fv \in \Gamma(M,V\times M)$. Therefore, by the universal property of the tensor product, we have a natural morphism of vector spaces $\phi:C^\infty(M) \otimes V \to \Gamma(M,V\times M)$.
Let $(v_i)_{i=1}^n$ be a basis of $V$. Then any section $s\in \Gamma(M,V\times M)$ can be written $s = \sum_{i=1}^n f_i v_i$ for some functions $f_i \in C^\infty(M)$. In other words, $\phi$ is surjective.
Conversely, any $T \in C^\infty(M)\otimes V$ can be written $T = \sum_{i=1}^n f_i\otimes v_i$ for some functions $f_i$. Then, if $\phi(T) = \sum_{i=1}^n f_iv_i =0$, we have $f_i = 0$ for $i=1,\ldots,n$, as $(v_i)$ is a basis of $V$. Therefore $\phi$ is injective.
Conclusion : $\phi$ is a natural isomorphism $C^\infty(M) \otimes V \simeq\Gamma(M,V\times M)$$