## Tag `0CLJ`

Chapter 86: Morphisms of Algebraic Stacks > Section 86.34: Valuative criteria

Lemma 86.34.9. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. Then the following are equivalent

- $f$ satisfies the uniqueness part of the valuative criterion,
- for every scheme $T$ and morphism $T \to \mathcal{Y}$ the morphism $\mathcal{X} \times_\mathcal{Y} T \to T$ satisfies the uniqueness part of the valuative criterion as a morphism of algebraic spaces.

Proof.Omitted. $\square$

The code snippet corresponding to this tag is a part of the file `stacks-morphisms.tex` and is located in lines 7835–7846 (see updates for more information).

```
\begin{lemma}
\label{lemma-uniqueness-representable}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks
which is representable by algebraic spaces. Then the following are equivalent
\begin{enumerate}
\item $f$ satisfies the uniqueness part of the valuative criterion,
\item for every scheme $T$ and morphism $T \to \mathcal{Y}$
the morphism $\mathcal{X} \times_\mathcal{Y} T \to T$ satisfies
the uniqueness part of the valuative criterion as a morphism
of algebraic spaces.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
```

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