Write $\mathcal{X} = [U/R]$ with $(U, R, s, t, c)$ as in Lemma \ref{lemma-well-nigh-affine}. Let $C$ be the ring of $R$-invariant functions on $U$, see Groupoids, Section \ref{groupoids-section-finite-flat}. We set $M = \Spec(C)$. The $R$-invariant morphism $U \to M$ corresponds to a morphism $f : \mathcal{X} \to M$ by Lemma \ref{lemma-quotient-compare}. The characterization of morphisms into affine schemes given in Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} immediately guarantees that $\phi : U \to M$ is a categorical quotient in the category of affine schemes. Hence $f$ is a categorical moduli space in the category of affine schemes (Lemma \ref{lemma-categorical-quotient-compare}). \medskip\noindent Since $\mathcal{X}$ is separated by Lemma \ref{lemma-well-nigh-affine} we find that $f$ is separated by Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-compose-after-separated}. \medskip\noindent Since $U \to \mathcal{X}$ is surjective and since $U \to M$ is quasi-compact, we see that $f$ is quasi-compact by Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-surjection-from-quasi-compact}. \medskip\noindent By Groupoids, Lemma \ref{groupoids-lemma-integral-over-invariants} the composition $$ U \to \mathcal{X} \to M $$ is an integral morphism of affine schemes. In particular, it is universally closed (Morphisms, Lemma \ref{morphisms-lemma-integral-universally-closed}). Since $U \to \mathcal{X}$ is surjective, it follows that $\mathcal{X} \to M$ is universally closed (Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-image-proper-is-proper}). To conclude that $\mathcal{X} \to M$ is a universal homeomorphism, it is enough to show that it is universally bijective, i.e., surjective and universally injective. \medskip\noindent We have $|\mathcal{X}| = |U|/|R|$ by Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-points-presentation}. Thus $|f|$ is surjective and even bijective by Groupoids, Lemma \ref{groupoids-lemma-points}. \medskip\noindent Let $C \to C'$ be a ring map. Let $(U', R', s', t', c')$ be the base change of $(U, R, s, t, c)$ by $M' = \Spec(C') \to M$. Setting $\mathcal{X}' = [U'/R']$, we observe that $M' \times_M \mathcal{X} = \mathcal{X}'$ by Quotients of Groupoids, Lemma \ref{groupoids-quotients-lemma-base-change-quotient-stack}. Let $C^1$ be the ring of $R'$-invariant functions on $U'$. Set $M^1 = \Spec(C^1)$ and consider the diagram $$ \xymatrix{ \mathcal{X}' \ar[d]^{f'} \ar[r] & \mathcal{X} \ar[dd]^f \\ M^1 \ar[d] \\ M' \ar[r] & M } $$ By Groupoids, Lemma \ref{groupoids-lemma-invariants-base-change} and Algebra, Lemma \ref{algebra-lemma-universally-bijective} the morphism $M^1 \to M'$ is a homeomorphism. On the other hand, the previous paragraph applied to $(U', R', s', t', c')$ shows that $|f'|$ is bijective. We conclude that $f$ induces a bijection on points after any base change by an affine scheme. Thus $f$ is universally injective by Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-universally-injective-local}. \medskip\noindent Finally, we still have to show that $f$ is a uniform moduli space in the category of affine schemes. This follows from the discussion above and the fact that if the ring map $C \to C'$ is flat, then $C' \to C^1$ is an isomorphism by Groupoids, Lemma \ref{groupoids-lemma-invariants-base-change}.