Maybe it is well known to experts or maybe it is just a stupid idea, but I will ask any way.

We know that if $X$ is a topological space, then there is an equivalence of categories between the category of locally constant sheaves (of sets) on $X$ and the category of covers (sous-entendu local homoemorphism) of $X$.

The equivalence is given by "$\Rightarrow$" using the "espace \'etal\'e" of sheaves, "$\Leftarrow$" taking the sheaf of sections.

Now I replace $X$ by a scheme (locally noetherien or something?), and I think there is an equivalence of categories between the category of locally constant constructible sheaves of sets (By constructible I mean the constant values should be finite) on the \'etale site of $X$ and the category of finite \'etale coverings of $X$.

I tried to construct the functor "$\Leftarrow$": Given $Y\rightarrow X$ finite \'etale, we associate to any $T\to X$, the set of sections $T\to Y\times_XT$. This is a locally constant constructible sheaf if $X$ is locally noetherian: You decompose $X$ into connect components and by SGA1 corollary 5.3 then you can see easily that on each connected component the association is a constant sheaf with a finite constant value.

Is this an equivalence? If it is how one constructs the quasi-inverse?

Furthermore, do we have any formulation like the finite representations of $\pi_1^{\text{et}}(X,x)$ is equivalent to the category of locally constant constructible \'etale sheaves (of vector spaces) on $X$. If this is true it should be a direct consequence of Grothendieck's main theorem on $\pi_1^{\text{et}}$ and the above statement.