Definition 101.39.10. 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 (101.39.1.1) and $\gamma $ as in Definition 101.39.1 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
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K') \ar[r] \ar@/^2em/[rr]_{x'} \ar[d]_{j'} & \mathop{\mathrm{Spec}}(K) \ar[d]_ j \ar[r]_-x & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A') \ar[r] \ar@/_2em/[rr]^{y'} & \mathop{\mathrm{Spec}}(A) \ar[r]^-y & \mathcal{Y} } \]
with induced $2$-arrow $\gamma ' : y' \circ j' \to f \circ x'$ is nonempty.
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