More details in the proof of the equivalence of categories between vector bundles over a smooth manifold and locally free sheaves $\def\VB{\mathsf{VB}}
\def\sO{\mathcal{O}}
\def\Mod{\mathsf{Mod}}
\def\LFMod{\mathsf{LFMod}}$For a ringed space $(X,\sO_X)$ and a ring $R$, denote $\Mod(\sO_X)$ and $\Mod(R)$ to the categories of $\sO_X$-modules and of $R$-modules. Let $M$ be a smooth manifold and denote $\sO_M$ to the sheaf of smooth functions on $M$. Denote $\VB(M)$ to the category of vector bundles over $M$ with vector bundle homomorphisms. For a vector bundle $\pi:E\to M$ over $M$, denote $\Sigma_E$ to the sheaf of sections of $\pi$. There is a functor
\begin{align*}
\Sigma:\VB(M)&\to\Mod(\sO_M)\\
E&\mapsto\Sigma_E\\
F:E\to E'&\mapsto F_*:\Sigma_E\to\Sigma_{E'},
\end{align*}
where $F_*$ is induced by postcomposition by $F$. (Actually, postcomposition induces a morphism of hom-sheaves $\mathcal{H}om_{\VB(M)}(E,E')\to\mathcal{H}om_{\sO_M}(\Sigma_E,\Sigma_{E'})$). Working in a local frame, one can see that $\Sigma$ is faithful by seeing that $\Gamma\circ\Sigma$ is faithful, where $\Gamma:\Mod(\sO_M)\to\Mod(\underbrace{\Gamma(M,\sO_M)}_{C^\infty(M)})$ denotes the global sections functor. Less trivial is that the functor $\Sigma$ is also full. For this, one first checks that the functor $\Gamma\circ \Sigma$ is full: this is Lemma 10.29 of Lee's Introduction to Smooth Manifolds. Leveraging Lee's proof, one can show after fullness of $\Sigma$: Given vector bundles $E$ and $E'$ over $M$ and an $\sO_M$-linear map $\varphi:\Sigma_E\to\Sigma_{E'}$, denote $F^U:E|_U\to E'|_U$ to the unique vector bundle homomorphism such that $\Gamma\circ\Sigma(F^U)=\Gamma(F^U_*)=\varphi_U$. Using the explicit definition of $F^U$ given in Lee's proof of Lemma 10.29, one can check that $F^U|_{U\cap V}=F^V|_{U\cap V}:E|_{U\cap V}\to E'|_{U\cap V}$. Therefore, the $F^U$'s glue to a vector bundle homomorphism $F:E\to E'$ such that $F|_U=F^U$. Thus, since $F_*|_U=(F|_U)_*=F_*^U$, one has $F_{*,U}=\Gamma(F^U_*)=\varphi_U$, i.e., $F_*=\varphi$.
This shows that $\Sigma$ is a fully faithful functor and hence an equivalence of categories into its essential image. It is clear that the image of $\Sigma$ is contained in the full subcategory of locally free $\sO_M$-modules. What I am interested is on finding a detailed proof that locally free $\sO_M$-modules are in fact the essential image of $\Sigma$. The only reference I am aware of is Ramanan's Global Calculus. He discusses this essential surjectivity in Chapter 2, between definitions 2.8 and 2.9. However, I find the argument sketched on his discussion a little bit unsatisfactory.
So my questions are:

*

*Do you know any reference which proves the essential surjectivity of $\Sigma:\VB(M)\to\LFMod(\sO_M)$ with more detail than Ramanan does?


*Do you know yourself a more detailed proof of this fact than Ramanan's one?
I was trying to combine the Vector Bundle Chart Lemma of Lee's book (Lemma 10.6 of the 2nd edition) with Ramanan sketch, but I wasn't sure how to actually do it.
 A: $\def\VB{\mathsf{VB}}
\def\sO{\mathcal{O}}
\def\Mod{\mathsf{Mod}}
\def\LFMod{\mathsf{LFMod}}
\def\frm{\mathfrak{m}}
\def\bbR{\mathbb{R}}
\def\ev{\operatorname{ev}}
\def\Ker{\operatorname{Ker}}
\def\grm{\operatorname{grm}}
\def\sE{\mathcal{E}}
\def\op{\oplus}
\def\bbK{\mathbb{K}}
\def\sF{\mathcal{F}}
\def\sG{\mathcal{G}}$The following proof is for real vector bundles over real smooth manifolds. However, I think that mutatis mutandis it works as well for complex vector bundles over real smooth manifolds (and maybe over complex manifolds too). But I haven't checked the details, so in the following everything is real in case of doubt. Anyhow, I will write $\bbK$ instead of $\bbR$ along the proof to suggest the possibility of generalization. (I would be grateful if someone points out a place in the literature where they do the complex generalization.)
Let $M$ be a smooth manifold and let $\sO_M$ be the sheaf of smooth functions on $M$. For each $p\in M$, denote $\frm_p\subset\sO_{M,p}$ to the (unique) maximal ideal of germs of smooth functions vanishing at $p$. The known result that evaluation at a point induces an isomorphism of vector spaces $\sO_{M,p}/\frm_p\cong\bbK$ generalizes to the following result:

Lemma 1. Let $\pi:E\to M$ be a vector bundle over $M$, and denote $E_p=\pi^{-1}(p)$ to the fiber at $p\in M$. Evaluation at a point induces a isomorphism of vector spaces
$$
\Sigma_{E,p}/\frm_p\Sigma_{E,p}\cong E_p,
$$
which in natural in $(E,\pi)$. The inverse of this map sends a vector $v\in E_p$ to a local section of $\pi$ around $p$ with value $v$ at $p$. (Actually, to the $\frm_p$-coset of the germ at $p$ of such a section.)

Proof. Fix $p\in M$. For an open neighborhood $U\subset M$ of $p$, the evaluation maps
\begin{align*}
\Sigma_E(U)&\to E_p\\
\sigma&\mapsto\sigma(p)
\end{align*}
make up a cocone over $E_p$. Hence, evaluation at $p$ induces a well-defined map $\ev_p:\Sigma_{E,p}\to E_p$. This map is easily seen to be onto by working at a local frame. It then suffices to see that $\Ker(\ev_p:\Sigma_{E,p}\to E_p)=\frm_p\Sigma_{E,p}$. The containment to the left is trivial. Conversely, let $\sigma\in \Sigma_{E,p}$ be in the kernel of $\ev_p$. Picking a local frame $(\sigma^i)$ around $p$, we get smooth functions $f_i$ defined locally around $p$ with $\sigma=\grm_p(f_i\sigma^i)$, where $\grm_p(-)$ denotes the germ at $p$. Then $0=\ev_p(\sigma)=f_i(p)\sigma^i(p)$. Hence $f_i(p)=0$ for all $i$. Thus $\sigma=\grm_p(f_i\sigma^i)=\grm_p(f_i)\grm_p(\sigma^i)\in\frm_p\Sigma_{E,p}$.
For the naturality, note that given a vector bundle homomorphism $F:E\to E'$, from the map on stalks of $F_*:\Sigma_E\to\Sigma_{E'}$ at $p$, we get an induced linear map $\overline{F}_{*,p}:\Sigma_{E,p}/\frm_p\Sigma_{E,p}\to\Sigma_{E',p}/\frm_p\Sigma_{E',p}$. On the other hand, $F$ induces a linear map $F_p:E_p\to E_p'$. Since the map $\overline{F}_{*,p}$ is given by postcomposition by $F$, we have $\ev_p\circ\overline{F}_{*,p}=F_p\circ\ev_p$. That is, the isomorphism is natural.
The proof of the last claim is immediate. Such a described section always exist by working in a local frame. $\square$

Lemma 2. Let $E$ and $E'$ be vector bundles over $M$ and let $\varphi:\Sigma_E\to\Sigma_{E'}$ be an $\sO_M$-linear map. The unique vector bundle homomorphism $F:E\to E'$ such that $F_*=\varphi$ is the one such that for each $p\in M$ the following diagram commutes:
$$
\require{AMScd}
\begin{CD}
E_p @>{F_p}>> E_p' \\
@V{\cong}VV @VV{\cong}V \\
\Sigma_{E,p}/\frm_p\Sigma_{E,p} @>{\overline{\varphi}_p}>> \Sigma_{E',p}/\frm_p\Sigma_{E',p} 
\end{CD}
$$
where $\overline{\varphi}_p$ is induced by the map on stalks $\varphi_p:\Sigma_{E,p}\to\Sigma_{E',p}$.

Proof. Let $v\in E_p$. By the proof of Lemma 10.29 of Lee's book, we have $F(v)=\varphi_M(\tilde{v})(p)$, where $\tilde{v}$ is a global section of $\pi$ with value $v$ at $p$. So
\begin{align*}
F_p(v)&=\varphi_M(\tilde{v})(p)\\
&=\ev_p(\varphi_M(\tilde{v}))\\
&=\ev_p(\grm_p(\varphi_M(\tilde{v})))\\
&=\ev_p(\varphi_p(\grm_p(\tilde{v})))\\
&=\ev_p(\varphi_p(\grm_p(\tilde{v}))+\frm_p\Sigma_{E',p})\\
&=\ev_p\circ\overline{\varphi}_p(\grm_p(\tilde{v})+\frm_p\Sigma_{E,p}).
\end{align*}
This expresses commutativity of the diagram if we traverse it as in
$
\begin{matrix}
\bullet&\rightarrow&\bullet\\
\downarrow&&\uparrow\\
\bullet&\rightarrow&\bullet
\end{matrix}
$. $\qquad\square$

Observation 3. For each $p\in M$, there is a functor into the category of $\bbK$-vector spaces
\begin{align*}
\mathsf{Mod}(\sO_M)&\to\mathsf{Vect}_\bbK\\
\sF&\mapsto\sF_p/\frm_p\sF_p\\
\varphi:\sF\to\sG&\mapsto\overline{\varphi}_p:\sF_p/\frm_p\sF_p\to\sG_p/\frm_p\sG_p
\end{align*}
The map $\overline{\varphi}_p$ is induced by the map on stalks $\varphi_p:\sF_p\to\sG_p$.

We are ready to show essential surjectiveness of the functor $\Sigma:\mathsf{VB}(M)\to\mathsf{LFMod}(\sO_M)$.

Proposition 4. Let $\sE$ be a locally free sheaf of $\sO_M$-modules of rank $r$. That is, each point in $M$ has an open neighborhood $U$ such that $\sE|_U\cong\sO_U^{\op r}$. Then there is a vector bundle $E$ over $M$ such that $\Sigma_E\cong\sE$ as $\sO_M$-modules.

Proof. The previous results impel us to define the vector space $E_p= \sE_p/\frm_p\sE_p$, for each $p\in M$. The dimension of this vector space is $r$, for
$$
\frac{\sE_p}{\frm_p\sE_p}\cong
\frac{\sO_{M,p}^{\op r}}{\frm_p\sO_{M,p}^{\op r}}
=\frac{\sO_{M,p}^{\op r}}{(\frm_p\sO_{M,p})^{\op r}}
=\frac{\sO_{M,p}^{\op r}}{\frm_p^{\op r}}
\cong\left(
\frac{\sO_{M,p}}{\frm_p}
\right)^{\op r}
\cong\bbK^r.
$$
We want to apply the vector bundle chart lemma (Lemma 10.6 of Lee's book) to the vector spaces $E_p$. For this, we will use the cover of $M$ of open subsets $U\subset M$ for which $\sE|_U$ is free of rank $r$.
Pick open subsets $U,V\subset M$ for which there are isomorphisms $\varphi:\sE|_U\cong\sO_U^{\op r}$ and $\psi:\sE|_V\cong\sO_V^{\op r}$. Consider the following diagram in the category of vector spaces:
$$
\require{AMScd}
\begin{CD}
E_p @>>> \{p\}\times\bbK^r \\
@| @AA{\cong}A \\
\sE_p/\frm_p\sE_p @>>> \sO_{M,p}^{\op r}/\frm_p^{\op r}
\end{CD}$$
where the isomorphism on the right is canonical and given by the component-wise evaluation at $p$. For each $p\in U$ (resp., for each $p\in V$) we define the map $\Phi_p$ (resp., the map $\Psi_p$) to be the unique map $E_p\to\{p\}\times\bbK^r$ for which the previous diagram commutes if the bottom map is $\overline{\varphi}_p$ (resp., $\overline{\psi}_p$).
Define $E=\bigsqcup_{p\in M}E_p$ and $\pi:E\to M$ to be the natural projection. Define also a pair of maps over $M$
\begin{align*}
\Phi&:\pi^{-1}(U)\to U\times\bbK^r \\
\Psi&:\pi^{-1}(V)\to V\times\bbK^r
\end{align*}
given by $\Phi|_{E_p}=\Phi_p$ for $p\in U$ and $\Psi|_{E_p}=\Psi_p$ for $p\in V$. To show that $(E,\pi)$ is a vector bundle over $M$, there is one thing left to verify from the statement of the Vector Bundle Chart Lemma. Namely, that the composite
$
\Psi\circ\Phi^{-1}|_{(U\cap V)\times\bbK^r}
$
is a bundle homomorphism.
Let $p\in U\cap V$ and consider the following commutative diagram:
$$
\require{AMScd}
\begin{CD}
(U\cap V)\times\bbK^r
@>{\Phi^{-1}}>> \pi^{-1}(U\cap V)
@>{\Psi}>> (U\cap V)\times\bbK^r \\
@AAA @AAA @AAA \\
\{p\}\times\bbK^r
@>{\Phi^{-1}_p}>> E_p
@>{\Psi_p}>> \{p\}\times\bbK^r \\
@A{\cong}AA @| @AA{\cong}A \\
\sO_{M,p}^{\op r}/\frm_p^{\op r}
@>{\overline{\varphi}^{-1}_p}>> \sE_p/\frm_p\sE_p
@>{\overline{\psi}_p}>> \sO_{M,p}^{\op r}/\frm_p^{\op r}
\end{CD}
$$
By functoriality of Observation 3, the bottom composite equals $(\overline{\psi\circ\varphi^{-1}})_p$. By Lemma 2, we have that $\Psi\circ\Phi^{-1}|_{(U\cap V)\times\bbK^r}$ is a bundle homomorphism. Namely, it is the unique automorphism of the trivial $r$-vector bundle over $U\cap V$ such that the induced morphism on the sheaf of sections equals $(\psi\circ\varphi^{-1})|_{U\cap V}$. (Recall that the sheaf of smooth sections of the trivial bundle $W\times \bbK^r\to W$ is canonically isomorphic to $\Sigma_{W\times \bbK^r}\cong\sO_{W}^{\op r}$.) In particular, $\Psi\circ\Phi^{-1}|_{(U\cap V)\times\bbK^r}$ is smooth and fiber-wise linear.
Hence, by the Vector Bundle Chart Lemma, $(E,\pi)$ is a vector bundle and $\Phi$ and $\Psi$ are local trivializations.
It remains to show that $\sE\cong\Sigma_E$. Let $U\subset M$ be an open subset such that there is an isomorphism $\varphi:\sE|_U\cong\sO_U^{\op r}$. As before, denote $\Phi:E|_U\to U\times\bbK^r$ to the associated trivialization of $(E,\pi)$. We have an isomorphism of $\sO_U$-sheaves
$$
\sE|_U
\xrightarrow{\varphi} \sO_U^{\op r}
\cong \Sigma_{U\times\bbK^r}
\xrightarrow{\Phi_*^{-1}|_U} \Sigma_E|_U.
$$
Abusing notation by identifying $\sO_U^{\op r}
\cong \Sigma_{U\times\bbK^r}$, denote $\Phi_*^{-1}|_U\circ\varphi$ to this composite isomorphism. Let's compute its value. Let $s\in\sE(U)$. Then
$$
(\Phi_{*,U}^{-1}\circ\varphi_U)(s)
=\Phi_{*,U}^{-1}(\varphi_U(s))
=\Phi^{-1}\circ(\varphi_U(s)).
$$
For $p\in U$, we have
\begin{align*}
(\Phi_{*,U}^{-1}\circ\varphi_U)(s)(p)
&=[\Phi^{-1}\circ(\varphi_U(s))](p)\\
&=\Phi^{-1}(\varphi_U(s)(p))\\
&=\Phi_p^{-1}(\varphi_U(s)(p))\\
&=\overline{\varphi}_p^{\,-1}(\grm_p(\varphi_U(s))+\frm_p^{\op r})\\
&=\varphi_p^{-1}(\grm_p(\varphi_U(s)))+\frm_p\sE_p\\
&=\grm_p(\varphi_U^{-1}\varphi_Us)+\frm_p\sE_p\\
&=\grm_ps+\frm_p\sE_p.
\end{align*}
This computation shows that $\Phi^{-1}_{*}\circ\varphi$ “does not depend” on the choice of isomorphism $\varphi:\sE_U\cong\sO_{U}^{\op r}$. More precisely: if $\psi:\sE_V\cong\sO_{V}^{\op r}$ is another isomorphism for another open $V\subset M$, the computation we just did proves that $(\Phi^{-1}_{*}\circ\varphi)|_{U\cap V}=(\Psi^{-1}_{*}\circ\psi)|_{U\cap V}$. Therefore, the map $\sE\to\Sigma_E$ which on sections at any open $U\subset M$ is given by
\begin{align*}
\sE(U)&\to\Sigma_E(U)\\
s&\mapsto U\to E\\
&\quad\;\;\; p\mapsto\grm_p s+\frm_p\sE_p
\end{align*}
is well-defined (i.e., $p\mapsto\grm_p s+\frm_p\sE_p$ is smooth), is $\sO_M$-linear and is an isomorphism, since locally it has the form $\Phi^{-1}_{*}\circ\varphi$. (We are using that for $\sF,\sG\in\Mod(\sO_M)$, the assingment $U\subset M\mapsto\operatorname{Hom}_{\sO_U}(\sF|_U,\sG|_U)$ is a sheaf and that “being an isomorphism” is a local condition for morphisms of sheaves.) $\quad\square$

EDIT: The previous proof is done by exploiting the natural s.e.s.
$$
0\to \frm_p\Sigma_{E,p}\to\Sigma_{E,p}\to E_p\to 0
$$
that exists for any smooth vector bundle $E$. On the other hand, I just learned that there is also another natural s.e.s.
$$
0\to\mu_p\Gamma(E)\to\Gamma(E)\to E_p\to 0
$$
where $\Gamma(E):=\Gamma(E,\Sigma_E)$ and $\mu_p:=\{f\in C^\infty(M)\mid f(p)=0\}$ is the maximal ideal of $C^\infty(M)$ of functions that vanish at $p$.
Thus, we get a natural isomorphism
$$
\tag{1}\label{1}
\Gamma(E)/\mu_p\Gamma(E)\cong E_p.
$$
I assume that the proof of the essential surjectiveness I gave here can be rewritten, mutatis mutandis, in terms of the isomorphism \eqref{1} instead of the one from Lemma 1 (e.g., one should rewrite the proof of Proposition 4 by defining $E_p=\Gamma(M,\sE)/\mu_p\Gamma(M,\sE)$ at the beginning). I think the main advantage of \eqref{1} is that it allows one to only work with cosets of sections, instead of with cosets of germs of sections, as we did above; as a result, the proof would use less cumbersome notation.
