The Stacks project

Lemma 83.5.6. Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \to U \times _ B U$ be a pre-relation over $B$. Then $j_\infty : R_\infty \to U \times _ B U$ is a pre-equivalence relation over $B$. Moreover

1. $\phi : U \to X$ is $R$-invariant if and only if it is $R_\infty $-invariant,

2. the canonical map of quotient sheaves $U/R \to U/R_\infty $ (see Groupoids in Spaces, Section 78.19) is an isomorphism,

3. weak $R$-orbits agree with weak $R_\infty $-orbits,

4. $R$-orbits agree with $R_\infty $-orbits,

5. if $s, t$ are locally of finite type, then $s_\infty $, $t_\infty $ are locally of finite type,

6. add more here as needed.