## Tag `0CLG`

Chapter 87: Morphisms of Algebraic Stacks > Section 87.38: Valuative criteria

Definition 87.38.6. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ satisfies the

uniqueness part of the valuative criterionif for every diagram (87.38.1.1) and $\gamma$ as in Definition 87.38.1 the category of dotted arrows is either empty or a setoid with exactly one isomorphism class.

The code snippet corresponding to this tag is a part of the file `stacks-morphisms.tex` and is located in lines 8752–8760 (see updates for more information).

```
\begin{definition}
\label{definition-uniqueness}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
We say $f$ satisfies the {\it uniqueness part of the valuative criterion}
if for every diagram (\ref{equation-diagram}) and $\gamma$
as in Definition \ref{definition-fill-in-diagram}
the category of dotted arrows is either empty or
a setoid with exactly one isomorphism class.
\end{definition}
```

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