The nature of the evidence is their writings, their experiences, their teachings, the witness of their lives… is that what you’re looking for, because you’re still not making a lot of sense, sorry.
Then you need to go read the ECFs, the didache, Tertullian, etc.
Are you asking us to post this stuff here?
The ECF’s spoke frequently of a unified church, and Peter’s primacy as the Pope (though the word pope wasn’t around yet, the office of the papacy and its primacy were clearly understood.)
Is this the sort of thing you’re looking for? If so, there’s way more of it than we could ever hope to post here. That’s why I’ve directed you to a couple of books that’ll hep you out.