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First, some notations and definitions:

1) For a vector space $V$:$$\mathsf{End}(V):=\{\mathsf{Linear}\ \mathsf{maps}\ f:V\longrightarrow V\}.$$ 2) A linear representation of a group $G$ is a pair $(V, \rho)$ consisting of a vector space $V$ and a map $\rho:G\longrightarrow \mathsf{End}(V)$ such that $$\rho_{gh}=\rho_g\circ \rho_h\quad \textrm{and}\quad \rho_{e}=\mathsf{id}_V,\quad\quad (\mathsf{Rep})$$ where $e$ is the identity of $G$.

3) A linear representation of a Lie groupoid $\mathsf{G}\rightrightarrows M$ is a pair $(E, \Delta)$ where $E\longrightarrow M$ is a vector bundle and $\Delta$ assigns to each morphism $g:x\longrightarrow y$, a linear isomorphism $\Delta_g:E_x\longrightarrow E_y$ such that $$\Delta_{gh}=\Delta_g\circ \Delta_h\quad \textrm{and}\quad \Delta_{1_x}=\mathsf{id}_{E_x},$$ where $1_x$ is the identity of $x$. Indeed, from $E$ we could define a Lie groupoid $\mathsf{Gl}(E)$ whose objects are points of $M$ and whose morphisms are linear isomorphisms $E_x\longrightarrow E_y$. Then, a linear representation of $\mathsf{G}$ is simply a functor $\Delta:\mathsf{G}\longrightarrow \mathsf{Gl}(E)$ covering the identity.

A linear representation of a group is the ''same'' as a functor $\rho:G\longrightarrow \mathsf{Vect}$ where $G$ is seen as the groupoid $G\rightrightarrows \{*\}$ and where $\mathsf{Vect}$ is the category of vector spaces. Taking this into account, is it possible to see a linear representation of a Lie grupoid $\mathsf{G}$ as a functor from $\mathsf{G}$ to the category of vector bundles?

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This is a partial answer. I thought the philosophy behind was more evolved but that is not the case. I guess the idea goes as follows:

Given $(E, \Delta)$ we associate the functor $\Delta^E:\mathsf{G}\longrightarrow \mathsf{Vect}$ which is given at the level of objects by $\Delta^E(x):=E_x$ and for a morphism $g:x\longrightarrow y$ of $\mathsf{G}$, $\Delta^E(g):=\Delta_g:E_x\longrightarrow E_y$.

On the other hand, given a functor $E:\mathsf{G}\longrightarrow \mathsf{Vect}$ we could assign $\bigsqcup_{x\in M} E(x)$. $\Delta$ is then obviously defined. However, I'm wondering: is there a way to ensure $\bigsqcup_{x\in M} E(x)$ is locally trivial?

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