Given a covering space $\varpi:Y \twoheadrightarrow X$ (with $X$ connected) if we consider the sheaf $\Gamma$ of sections of this bundle we can show that $\Gamma$ is a local system. I would like to know if there is an equivalence or at least a pair of adjoint functors between the category of local systems on $X$ and the category of covering spaces of $X$. I tried to see if étalé space does the trick with a constant sheaf but I don't know if it is right.
If we start with a constant sheaf $\mathcal{L}$ with stalk $L$ and apply the étalé space construction we obtain a bundle $\pi:\Lambda \mathcal{L} \twoheadrightarrow X$. Since $X$ is connected locally constant functions are exactly the constant functions. Let $l \in L$ and consider $X_l=\{ \operatorname{germ}_p l \text{ | }p \in X \}$. Clearly $X_l$ is an open subset of $\Lambda \mathcal{L}$ such that $\pi|_{X_l}$ is an homeomorphism $X_l \cong X$ and $\Lambda \mathcal{L}=\bigsqcup X_l$. So we get a trivial covering space.
Is this going to work for the general case of local systems? Covering spaces are very related to $\pi_1(X,p)$, does this affect the possible equivalence of categories?