Fundamental result on the projective tensor product of sections of a vector bundle Definition Let $E_{1} \xrightarrow{\pi_{1}} M_{1}$, $E_{2} \xrightarrow{\pi_{2}} M_{2}$ be two vector bundles over $M_{1}$ and $M_{2}$ with fibers $V_{1}$, $V_{2}$ respectively. The exterior tensor product $E_{1} \boxtimes E_{2}$ is defined as the vector bundle over $M_{1} \times M_{2}$, whose fiber over $(x, y) \in M_{1} \times M_{2}$ is $E_{1 x} \otimes E_{2 y}$, where $E_{1 x}$ is the fibre of $E_{1}$ over $x$ and $E_{2 y}$ is the fibre of $E_{2}$ over $y$.
In the article Brouder, Dang, Laurent-Gengoux, and Rejzner - Properties of field functionals and
characterization of local functionals they have the so called fundamental result on the projective tensor product of sections of a vector bundle

Proposition III.8. Let $\Gamma(M, B)$ be the space of smooth sections
of some smooth finite rank vector bundle $B \rightarrow M$ on a
manifold $M$. Then $\Gamma(M, B)^{\hat{\otimes}_{\pi}
 k}=\Gamma\left(M^{k}, B^{\boxtimes k}\right)$.

Where $\boxtimes$ means exterior tensor product and $\otimes_\pi$  is the projective tensor product.
I am trying to prove this theorem. My first step is to prove the algebraic isomorphism.
This is how  I am trying to prove the algebraic isomorphism by following the same steps as in A nonlinear theory of generalized tensor fields
on Riemannian manifolds by Eduard Nigsch at page 89:
Let  $V$ be the fiber of  the trivial vector bundle $B \xrightarrow{\pi} M$.
We know there is a  bilinear map from $\alpha :V\times V \rightarrow V\otimes V$ with $$\alpha(v_1,v_2)=v_1 \otimes v_2$$
where $v_1,v_2 \in V$.
Now let $\psi :B \rightarrow U\times V$ be a local trivialization map with $U \subset M$.
Now we define the local sections $\sigma_i$ by
$\sigma_i(x) =\psi^{-1}(x,v_i)$.
We also define the local sections $\gamma_{ij}$ by
$\gamma_{ij} =\phi^{-1}(x,y,v_i\otimes v_j)$
where $\phi :B{\boxtimes}B \rightarrow U^2 \times V\otimes V$ is a local trivialization map  of the bundle $B{\boxtimes}B \rightarrow M \times M$.
Define the map
$g :\Gamma\left(M, B\right) \times \Gamma\left(M, B\right) \rightarrow \Gamma\left(M\times M, B{\boxtimes}B\right)$ by
$$g=\sum_{i, j} m \circ\left(i d \times m\left(\cdot, \gamma_{i j}\right)\right) \circ\left(\sigma_{i}^{*} \times \sigma_{j}^{*}\right)$$
where  $m: C^{\infty}(M) \times \Gamma(B\otimes B)\rightarrow \Gamma(B\otimes B) $ is the module multiplication on $\Gamma(B\otimes B)$  and  $\sigma_{i}^{*}$ is the dual  of $\sigma_{i}$.
By the properties  of the tensor product  there is a  bilinear map $f: \Gamma(M, B)\otimes \Gamma(M, B)\rightarrow \Gamma\left(M\times M, B{\boxtimes}B\right)$  such that if $g$ is the map $h:\Gamma\left(M, B\right) \times \Gamma\left(M, B\right) \rightarrow \Gamma(M, B)\otimes \Gamma(M, B)$
we have that $$g=f \circ h.$$
For the inverse define $h^{-1}: \Gamma\left(M\times M, B{\boxtimes}B\right)\rightarrow \Gamma(M, B)\otimes \Gamma(M, B)$
$h^{-1}(s)=\sum_{i, j} \gamma_{i j}^{*}(s) \alpha_{i} \otimes \beta_{j}$ for $s \in \Gamma\left(M\times M, B{\boxtimes}B\right)$
Now we have that $$
\begin{aligned}
&h^{-1}\left(f\left(t \otimes u\right)\right)=h^{-1}\left(h  \left(t^{i} \alpha_{i}, u^{j} \beta_{j}\right)\right)=h^{-1} \left(t^{i} u^{j} \gamma_{i j}\right)=t^{i} u^{j} \alpha_{i} \otimes \beta_{j}\\
&=t^{i} \alpha_{i} \otimes u^{j} \beta_{j}=t \otimes u \text { and }\\
&h(h^{-1}(s))=h\left(s^{i j} \alpha_{i} \otimes \beta_{j}\right)=s^{i j} g\left(\alpha_{i}, \beta_{j}\right)=s^{i j} \gamma_{i j}=s .
\end{aligned}
$$
Is my proof correct?
 A: The tensor product (either one) is symmetric $A \otimes B \cong B \otimes A$, associative $(A\otimes B) \otimes C \cong A \otimes (B \otimes C)$ and distributive over direct sums $A \otimes (B \oplus C) \cong (A\otimes B) \oplus (A\otimes C)$. Spaces of sections preserve direct sum decompositions $\Gamma(M, E_1 \oplus E_2) \cong \Gamma(M,E_1) \oplus \Gamma(M,E_2)$. Any vector bundle $E\to M$ can be realized as a summand of a trivial vector bundle $(V\times M \to M) \cong (E\oplus E' \to M)$, where we can realize the sub-bundles $E$ and $E'$ and the image and kernel of a fiber-wise projection $P$ on $V\times M$ (it is sufficient that $M$ is contractible to a compact space). The completed tensor product satisfies the identity $C^\infty(M_1) \otimes C^\infty(M_2) \cong C^\infty(M_1 \times M_2)$ (for details see Trèves opological Vector Spaces, Distributions and Kernels 1970).
With the above background, the proof is straightforward. Start with the observation that for a trivial bundle $\Gamma(M, V\times M) \cong C^\infty(M) \otimes V$. Hence $$\Gamma(M_1,V_1\times M_1) \otimes \Gamma(M_2,V_2\times M_2)\cong C^\infty(M_1\times M_2) \otimes (V_1\otimes V_2) \cong \Gamma(M_1\times M_2, (V_1 \times M_1) \boxtimes (V_2 \times M_2)).$$
Applying direct sum decompositions to both sides gives
$$
  (\Gamma(M_1,E_1) \oplus \Gamma(M_1,E'_1)) \otimes (\Gamma(M_2,E_2) \oplus \Gamma(M_2,E'_2))
  \cong \Gamma(M_1\times M_2, (E_1\oplus E'_1) \boxtimes (E_2\oplus E'_2)) .
$$
Finally, the desired isomorphism
$$
  \Gamma(M_1,E_1) \otimes \Gamma(M_2,E_2)
  \cong \Gamma(M_1\times M_2, E_1 \boxtimes E_2)
$$
follows because both sides coincide with the image of the projection $P_1\otimes P_2$ (interpreted as acting fiber-wise or on spaces of sections as needed).
