TITLE: locally constant constructible sheaves and finite etale coverings
QUESTION [10 upvotes]: 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.

REPLY [3 votes]: To construct a quasi-inverse, you may use the equivalence of categories between affine morphisms and sheaves of quasicoherent algebras described in EGA2 Chapter 1.  Given a locally constant constructible sheaf $F$ of sets, you can take the étale sheafification of the presheaf of algebras $U \mapsto \mathscr{O}_U^{F(U)}$ on the small étale site of $X$.  The relative spectrum of this sheaf is the finite étale cover you want.  It looks like you need some descent to prove this, so this construction is more or less a disguised version of Angelo's.
If you have a pointed connected scheme $(X,x)$, then there is an equivalence between finite representations of the fundamental group on a vector space, and locally constant constructible étale sheaves of vector spaces.  In one direction, a sheaf $F$ is taken to the fiber over $x$.  In the other direction, you pass to a trivializing cover, take a constant sheaf of appropriate dimension, and apply the associated sheaf construction (which is just descent).  If your scheme is not connected, you replace the fundamental group with the fundamental groupoid.