Let sheaf $\mathcal{F}$ on $X$ be a local system, ie. for every $x$ in $X$, there is an an open set $U$ containing $x$ such that $\left.\mathcal{F}\right|_U$ is isomorphic to a constant sheaf $\underline{V}_X$ associated with some finite-dimensional vector space $V$ over field $\mathbb{K}$. (We assume that $X$ is a nice topological space.) I know that the category of local systems - which I denote by $\text{Loc}(X,\text{Vect}_{\mathbb{K}}^{\text{fin}})$ - is a full subcategory of the category of all sheaves of vector spaces. I am trying to understand the monodromy representation functor.
What is the stalk $\mathcal{F}_x$ of sheaf $\mathcal{F}$ at any point $x$ in $X$? I know that the stalk is defined to be the direct limit of sections $\mathcal{F}(U)$ over open sets $U$ containing $x$, but I don’t really understand how to determine such an abstract object.
How can I show that the rule $\mathcal{F}\mapsto\mathcal{F}_x$ for all $x$ in $X$ extends to the functor $$\text{mon}:\text{Loc}(X,\text{Vect}_{\mathbb{K}}^{\text{fin}})\longrightarrow\text{Fun}(\Pi_1(X), \text{Vect}_{\mathbb{K}}^{\text{fin}})$$ from the category of locally constant sheafs to the category of functors between the fundamental groupoid $\Pi_1(X)$ and finite-dimensional vector spaces $\text{Vect}_{\mathbb{K}}^{\text{fin}} $? I know that it will have to be defined on objects and on morphisms.
I have read that the $\text{mon}$-functor induces an equivalence of categories. I know that this is the case as soon as it is fully faithful and essentially surjective, but how can I prove this? Moreover, I was wondering which type of object on the left hand side corresponds to constant sheaves, but I don’t really get a feeling for this monodromy representation...
I am happy for your help!
