I came across with the following question

Definition of "descends to"

I think that is a question that many students of algebraic geometry (and probably many others) have in the beginning, and I don't know any formal reference for the question. Although the accepted answer gives an idea of what it means to say that a map descents to another, I don't think this is enough, I would like to see a formal definition or a reference.

Because of this, I would like to know if I can start a bounty on this question.