Lemma 69.22.3. Let $S$ be a scheme. Let $f : X \to Y$ and $h : U \to X$ be morphisms of algebraic spaces over $S$. Assume that $Y$ is locally Noetherian, that $f$ and $h$ are of finite type, that $f$ is quasi-separated, and that $h(U)$ is dense in $X$. If given any commutative solid diagram
where $A$ is a discrete valuation ring with field of fractions $K$, there exists a unique dotted arrow making the diagram commute, then $f$ is proper.