Properties preserved by categorical quotient (after Mumford's GIT)

Question

A categorical quotient of a variety $X$ acted by $G$ is a morphism $f: X \to Y$ constant on $G$-orbits such that every $h: X \to Z$ constant on $G$-orbits factors through it.

Mumford on the page 5 of Geometric Invariant Theory states that if $X$ is

1. reduced,
2. connected,
3. irreducible,
4. locally integral,
5. or locally integral and normal,

then $Y=X//G$ also is. Is it really obvious from the universal property (and why)?

Answer 1

First, let me make your statement a bit more precise. Saying the map factors through it (in your first line), while correct, should be there exists a unique morphism $g:Y\to Z$ etc.

Now, let me do this for 1. If $Y$ is not reduced, then let $Z=Y_{red}$. So, Then it is clear that $f$ factors through $Z$ and then you have $h:X\to Z$ and thus $g:Y\to Z$. Now, we have two maps $Id:Y\to Y, g:Y\to Z\subset Y$, and so by uniqueness, $Id=g$ as morphisms and then easy to see that $Y=Z$.