Definition 100.39.10. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ satisfies the *existence part of the valuative criterion* if for every diagram (100.39.1.1) and $\gamma $ as in Definition 100.39.1 there exists an extension $K'/K$ of fields, a valuation ring $A' \subset K'$ dominating $A$ such that the category of dotted arrows for the outer rectangle of the diagram

with induced $2$-arrow $\gamma ' : y' \circ j' \to f \circ x'$ is nonempty.

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