This is cross-posted from `math.se`

after receiving points and no answers. I apologise if this question is too basic for MathOverflow.

I'm refreshing my memory of covering space theory, and this time around, I know some sheaf theory. It feels like arguments are used to prove results about covering spaces, such as uniqueness of lifts, having something "sheafy" about them.

For example, to prove uniqueness of lifts, we argue by trying to extend "equality at a point" to "equality over a neighbourhood" to "equality over the entire domain". It seems like the language of sheaves may make this clearer?

Similarly, when it comes to covering spaces, there is something "etale-like" about them. Is there a reference that expands on this perspective?