Let $X$ be a nice topological space (locally path-connected and simply connected). Assume also that $X$ is connected and fix a base point $x_0 \in X$. We consider the category $Cov(X)$ of covering spaces of $X$, and the fiber functor $F : Cov (X) \to Sets$ sending $\pi : E \to X$ to $\pi^{-1} (x_0)$. Is there an organized reference detailing the proof that $F$ is a monadic functor, without first developing explicitly the equivalence with sets acted upon by the fundamental group or some other equivalence, but working directly (so in the spirit of Grothendieck's definition of the etale fundamental group)?
Thanks