The seminal book by Mumford titled "Geometric Invariant Theory", 1965, gave a thorough and satisfactory answer to the question of how to form quotients of schemes. Not only that, it laid the foundations of a whole new subfield of algebraic geometry, aptly named geometric invariant theory, that has had numerous applications in birational geometry, moduli spaces, symplectic geometry, Kähler geometry, algebraic spaces, stacks and coding theory, to name a few.
The modern reference for geometric invariant theory is still the original work of Mumford, now in its third edition: Mumford, Fogarty, Kirwan - Geometric Invariant Theory, 1994. One of the basic definitions is that of a "geometric quotient", which is given in Definition 0.6 of the book. Unfortunately, it seems to contain an error. Namely, it contains the following sentence under item (iii):
This is without any prior introduction of the symbols $U'$, $Y'$, $X'$ or the strange 'C' symbol, which is not the same as the subset symbol which appears right before. The line looks very out of place as if it shouldn't be there. What is worse, it also appears in the exact same way in the second edition of the book. Am I missing something or should that line be deleted?
For reference, the whole definition:
Yes, that line should be deleted.
There is a blog post from 2010 by Aise Johan de Jong, the maintainer of the Stacks Project, where in the comments they come to the conclusion that in the first edition of Mumford's book, geometric quotient was defined to be universally submersive in Definition 0.6 (iii), but Fogarty had changed that to just submersive from the second edition onwards, but an unfortunate typesetting error still left the aformentioned line with $U'$ and $Y'$.
Note that the definition of geometric quotient is different from paper to paper, some requiring universally submersive, some only submersive.