Especially when taking quotients by group actions the following definition is useful.
Definition 83.9.1. Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \to U \times _ B U$ be a pre-relation. A morphism $\phi : U \to X$ of algebraic spaces over $B$ is called a good quotient if
$\phi $ is invariant,
$\phi $ is affine,
$\phi $ is surjective,
condition (3) holds universally, and
the functions on $X$ are the $R$-invariant functions on $U$.
In [seshadri_quotients] Seshadri gives almost the same definition, except that instead of (4) he simply requires the condition (3) to hold – he does not require it to hold universally.
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