I found this paper on van Kampen's theorem (https://www3.nd.edu/~andyp/notes/SeifertVanKampen.pdf), and I was wondering how to prove Lemma 1:
Let Z be a reasonable nonempty path-connected space, let G be a group, and let p ∈ Z. Then there is a natural bijection
{homomorphisms π_1(Z, p) → G} ↔ {based regular G-covers (Y, q) → (Z, p)}.
Why is this true? This is certainly not immediately clear to me. Also, how does the version of van Kampen's theorem in this paper relate to the version using amalgamated free products?
Note that I'm only self-taught via Hatcher's book, so I apologize if this is obvious.