What is the relation between $C^\infty$-linear and tensorial? I've been self studying Riemannian Geometry through Spivak's and Lee's books, and fairly often I've seen an argument that goes somewhat like this:
We have an operator that acts on vectors in the tangent space and, in order to show that it "lives at points", we generalise it to operate on global sections in vector bundles. Then, by showing this operator is linear over $C^\infty$ functions, some argument involving bump functions implies the operator defines a tensor field. 
Usually this is followed by some remark about modules, giving a strong impression that something bigger is going on.
On general relativity texts, there's an equivalent practice that involves showing the operator "transforms correctly".
This is all quite vague, since I cannot quite point my finger at what's the general phenomenon. So my question is:
What is the relation between tensor fields and $C^\infty$ linearity and what does this has to do with modules?
 A: First note that every vector bundle $E$ gives you $C^{\infty}(M)$-Module structure on global sections $\Gamma(E).$
Additionaly for vector bundles $E,F$ on maniflod $M$ there are $C^{\infty}(M)$-Module isomorphism
$$\Gamma(E)\otimes_{C^{\infty}(M)}\Gamma(F)\simeq\Gamma(E\otimes_{\mathbb{R}} F)\hspace{10pt}\text{and}\hspace{10pt}\Gamma(E^{*})\simeq\Gamma(E)^{\vee}$$
where $E\otimes_{\mathbb{R}} F$ is bundle with tensor product taken fiberwise and $E^*$ is bundle with dual vector space taken fiberwise.
Now any tensor field is a section of bundle $TM\otimes\dots\otimes TM\otimes T^*M\dots\otimes T^*M.$ From previous you get that any vetor field can be treated as element of
$$\Gamma(TM)\otimes\dots\otimes \Gamma(TM)\otimes \Gamma(TM)^{\vee}\dots\otimes \Gamma(TM)^{\vee}.$$
There are also isomorphism for symmetric and screw symmetric tensors. Namely
$$\Gamma(E\odot E)\simeq\Gamma(E)\odot_{C^{\infty}(M)}\Gamma(E)\hspace{5pt}\text{and}\hspace{5pt}\Gamma(E\wedge E)\simeq\Gamma(E)\wedge_{C^{\infty}(M)}\Gamma(E).$$
Last, but not least due to the famous Swan's theorem you have that for any vector bundle $E$
$$\Gamma(E)^{\vee\vee}\simeq\Gamma(E).$$
It allows you to treat ${C^{\infty}(M)}$ linear functionals on $\Gamma(TM)^{\vee}$ as elements of $\Gamma(TM).$
For reference see:
Jet Nestruev, Smooth manifolds and observables, 2003
Lawrence Conlon, Differentiable Manifolds SE, 2001 
A: $\def\homb{\mathcal{H}om}
\def\bbR{\mathbb{R}}
\def\hom{\operatorname{Hom}}
\def\shom{\operatorname{SHom}}
\def\VB{\mathsf{VB}}
\def\sO{\mathcal{O}}
\def\sT{\mathcal{T}}
\def\mult{\operatorname{Mult}}
\def\multb{\mathcal{M}ult}
\def\smult{\operatorname{SMult}}
$One could show the result succintly using only universal properties and natural isomorphisms, yes. But doing so requires a somewhat heavy usage of the theory of smooth vector bundles. I will explain here a summary of the relevant facts that are used in the concise proof given at the end.
For any vector bundles $\pi:E\to M$ and $\pi':E'\to M$ over $M$, the hom-bundle of $E$ and $E'$, denoted $\homb(E,E')$, is the vector bundle over $M$ whose fiber over $p\in M$ equals $\homb(E,E')_p=\hom(E_p,E'_p)$. Denote $\homb(\pi,\pi'):\homb(E,E')\to M$ to the projection. The smooth structure on $\homb(E,E')$ is defined as follows: given local trivializations of $E$ and $E'$ over $U\subset M$,
\begin{align*}
\Phi:\pi^{-1}(U)&\to U\times\bbR^k,
\\v\in E_p&\mapsto(p,\varphi_pv),\\\\
\Phi':(\pi')^{-1}(U)&\to U\times\bbR^{k'},
\\v\in E_p'&\mapsto(p,\varphi'_pv),
\end{align*}
one can define the candidate for trivialization
\begin{align*}
\homb(\Phi,\Phi'):\homb(\pi,\pi')^{-1}(U)&\to U\times\bbR^{k\times k'}\\
A\in\hom(E_p,E_p)&\mapsto(p,\varphi'_pA\varphi_p^{-1})
\end{align*}
and use the Vector Bundle Chart Lemma (Lemma 10.6 of Lee's Introduction to Smooth Manifolds) to prove that the maps $\homb(\Phi,\Phi')$ are trivializations of a unique smooth vector bundle structure on $\homb(E,E')$.
In particular, given a vector bundle $E$ over $M$, one has $E^*=\homb(E,M\times\bbR)$.
Using a similar approach, one could define the tensor product of the bundles $E$ and $E'$ to be a bundle whose fiber over $p$ is $E_p\otimes_\bbR E_p'$. Following a similar strategy as before, endow the set $E\otimes E'=\bigsqcup_{p\in M}E_p\otimes_\bbR E_p'$ with a smooth structure that turns the projection $\pi\otimes\pi':E\otimes E'\to M$ into a smooth vector bundle.
Denote now $\hom_{\VB(M)}(E,E')$ to the set of vector bundle homomorphisms $E\to E'$. You can verify that the assignment $U\subset M \mapsto \hom_{\VB(U)}(E|_U,E'|_U)$ is sheaf. Actually, a sheaf of $\sO_M$-modules (where $\sO_M$ is the sheaf of smooth real-valued functions on $M$), the hom-sheaf, that we denote as $\shom_{\sO_M}(E,E')$ or just $\shom(E,E')$. Given a vector bundle $E$, denote $\Sigma_E$ to the sheaf of sections of $E$, which is a sheaf of $\sO_M$-modules.

Exercise 1. Show that the map of sheaves
$$
\Sigma_{\homb(E,E')}\to\shom(E,E')
$$
which on sections over $U\subset M$ is given by
\begin{align*}
\Sigma_{\homb(E,E')}(U)&\to\hom_{\VB(U)}(E|_U,E'|_U)\\
\sigma&\mapsto E|_U\to E'|_U\\
& \:\, v\in E_p\mapsto\sigma(p)v
\end{align*}
is an isomorphism of $\sO_M$-modules.


Exercise 2. Show that the map
$$
E_1^*\otimes\dots\otimes E_k^*\to(E_1\otimes\dots\otimes E_k)^*
$$
which fiberwise is given by the natural isomorphism of vector spaces
$$
E_{1,p}^*\otimes\dots\otimes E_{k,p}^*\to(E_{1,p}\otimes E_{k,p})^*
$$
is an isomorphism of vector bundles.


Remark 3. The global sections functor, from the category of smooth vector bundles over $M$ to the category of $C^\infty(M)$, is fully faithful. See Theorem 12.29 of J. Nestruev, Smooth Manifolds and Observables.


Remark 4. Given vector bundles $E$ and $E'$ over $M$, one has $\Gamma(E\otimes E')\cong\Gamma(E)\otimes_{C^\infty(M)}\Gamma(E')$. This is non-trivial and requires the usage of the Serre-Swan theorem. See Theorem 12.39 of J. Nestruev's book.

Now, note that
\begin{align*}
T^kT^*M&=T^k(TM)^*
&\text{by definition,}
\\
&\cong (T^kTM)^*
&\text{by Exercise 2,}
\\
&=\homb(T^kTM,M\times\bbR).
\end{align*}
Thus, abbreviating $\sT^k(M)=\Gamma(T^kT^*M)$, we have
\begin{align*}
\sT^k(M)
&\cong\hom_{\VB(M)}(T^kTM,M\times\bbR)
&\text{by Exercise 1,}
\\
&\cong\hom_{C^\infty(M)}(\Gamma(T^kTM),C^\infty(M))
&\text{by Remark 3,}
\\
&\cong\hom_{C^\infty(M)}(T^k\underbrace{\Gamma(TM)}_{\mathfrak{X}(M)},C^\infty(M))
&\text{by Remark 4,}
\\
&\cong\mult_{C^\infty(M)}(\mathfrak{X}(M),k;C^\infty(M)),
\end{align*}
where in the last step we have used the universal property of the tensor product: here $\operatorname{Mult}_{C^\infty(M)}(\mathfrak{X}(M),k;C^\infty(M))$ is the $C^\infty(M)$-module of $C^\infty(M)$-multilinear maps $\prod_{i=1}^k\mathfrak{X}(M)\to C^\infty(M)$.

EDIT: Here is another proof idea that avoids the usage of Remark 4, possibly the most non-trivial result of all the ones we invoked.
Let $E_1,\dots,E_k,B$ be vector bundles over $M$. Define a multilinear homomorphism of vector bundles to be a smooth map $E_1\oplus\cdots\oplus E_k\to B$ that is fiberwise multilinear, where $\oplus$ denotes the Whitney sum of vector bundles. Denote $\mult_{\VB(M)}(E_1,\dots,E_k;B)$ to be the $C^\infty(M)$-module of multilinear homomorphisms of vector bundles $E_1\oplus\cdots\oplus E_k\to B$.
Remarks 5.

*

*Show that the canonical map $E_1\oplus\cdots\oplus E_k\to E_1\otimes\cdots\otimes E_k$ is a multilinear homomorphism of vector bundles. Prove the universal property of the tensor product of vector bundles. Namely, that the map
$$
\hom_{\VB(M)}(E_1\otimes\cdots\otimes E_n,B)\to\mult_{\VB(M)}(E_1,\dots,E_n,B)
$$
given by precomposition with $E_1\oplus\cdots\oplus E_k\to E_1\otimes\cdots\otimes E_k$ is an isomorphism of $C^\infty(M)$-modules.


*Show that the global sections functor induces a natural isomorphism
$$
\mult_{\VB(M)}(E_1,\dots,E_k;B)\cong\mult_{C^{\infty}(M)}(\Gamma(E_1),\dots,\Gamma(E_k);\Gamma(B)).
$$
This can be done by trying to generalize the proof of Lemma 12.24 or of Lemma 10.29 of J. M. Lee Introduction to Smooth Manifolds, 2nd ed.
So we have
\begin{align*}
E_1^*\otimes\cdots\otimes E_k^*
&\cong(E_1\otimes\cdots\otimes E_k)^*
&\text{by Exercise 2,}
\\
&=\homb(E_1\otimes\cdots\otimes E_n,M\times\bbR).
\end{align*}
Therefore,
\begin{align*}
\Gamma(E_1^*\otimes\cdots\otimes E_k^*)
&\cong\hom_{\VB(M)}(E_1\otimes\cdots\otimes E_k;M\times\bbR)
&\text{by Exercise 1,}
\\
&\cong\mult_{\VB(M)}(E_1,\dots,E_k;M\times\bbR)
&\text{by Remark 5.1,}
\\
&\cong\mult_{C^\infty(M)}(\Gamma(E_1),\dots,\Gamma(E_k);C^\infty(M)).
&\text{by Remark 5.2.}
\end{align*}
Particularizing to the case $E_i=T^*M$ for all $i$ gives the desired result. This proof is essentially the same as Lee's one given in Lemma 12.24 of his smooth manifolds book. The difference is that here we give the full picture.
