Definition 26.20.3. Let $f : X \to S$ be a morphism of schemes. We say $f$ satisfies the existence part of the valuative criterion if given any commutative solid diagram
where $A$ is a valuation ring with field of fractions $K$, the dotted arrow exists. We say $f$ satisfies the uniqueness part of the valuative criterion if there is at most one dotted arrow given any diagram as above (without requiring existence of course).