An equivalence of categories Let $F: \Pi(X) \to \text{SET}$ be a functor, where $\Pi(X)$ is the fundamental groupoid of $X$. I have shown earlier that we can construct a covering $p: Y = \bigsqcup_{x\in X} F(x) \to X$ from the functor $F$. This covering takes $F(x)$ to $x$.
Now let $\mathcal{C}_F$ be the category whose objects are pairs $(x,y)$ with $x\in X$ and $y\in F(x)$. A morphism $(x,y)\to (x',y')$ is a homotopy class $[\gamma]$ where $\gamma$ is a path from $x$ to $x'$ and $F(\gamma)(y) = y'$.
I would like to show the categorical equivalence of the fundamental groupoid of $Y = \bigsqcup_{x\in X} F(x)$ and $\mathcal{C}_F$.
My attempt:
We need to find two functors. Let's begin with a funcor $\Pi(Y) \to \mathcal{C}_F$. We assign to an object $y \in Y$ the object $(x,y)\in \text{ob}\mathcal{C}_F$ where $y\in F(x)$.
To a morphism from $y$ to $y'$, i.e. a path class $[\alpha]$ from $y$ to $y'$, we assign the path class $[p \circ \alpha]$ from $x$ to $x'$.
Now let's write down a functor $\mathcal{C}_F \to \Pi(Y)$. We simply send the object $(x,y)$ to $y \in Y$.
What do we do with a morphism $(x,y) \to (x',y')$, i.e. a path class $[\gamma]$ from $x$ to $x'$ with $F(\gamma)(y) = y'$?
 A: If $G$ is a groupoid, there is a simple but useful equivalence between 


*

*The category of actions of $G$ on sets, i.e. of functors $G \to Sets$, and

*The category of groupoid covering morphisms of the groupoid $G$. 
Note that a covering morphism  $q: H \to G$ of groupoids is a morphism such that for each $y \in Ob(H)$ and $g \in G$ starting at $q(y)$ there is unique $h \in H$ starting at $y$ such that $q(h)=g$. 
Now your question rephrases as: given a covering morphism $q: H \to \pi_1(X)$, how might we construct a covering space $Y$ of $X$? Clearly we want to take the set $Y$ to be $Ob(H)$. To get the topology on $Y$ you need a local condition on $X$: this is that for any $y \in Y$ there exists a path connected neighbourhood $N$ of $q(y)$ in $X$ such that the morphism $\pi_1(N) \to \pi_1(X) $ lifts uniquely to a morphism $\phi: \pi_1(N) \to H$ such that $\phi(q(y))=y$. The details are in Topology and Groupoids, Chapter 10. 
The advantage of this approach to that of actions is that liftings are more easily described in terms of covering morphisms than in terms of actions. 
This illustrates a general principle in mathematics:  two notions may be equivalent, even in a categorical sense, but where you should operate for a particular situation may be easier to understand in one notion than in the other. The more difficult the equivalence between the two notions, the more advantage there is in being  able to move from one to the other according to convenience. 
