Most general lifting property for proper morphisms

Question

Let $\mathcal C$ be the class of morphisms $f\colon U\to V$ of schemes such that for every proper map $g\colon X\to Y$ between schemes and every commutative solid square

there exists a lift $h$ making the diagram commute.

Is there a nice description of elements of $\mathcal C$? Or at least of some subclass of $\mathcal C$ beyond the morphisms used in the valuative criterion, where $V$ is the spectrum of a valuation ring and $U$ is the spectrum of its fraction field? I am particularly interested in the case where $V=\mathbb A^1\times S$ and $U=(\mathbb A^1\setminus 0)\times S$, for any scheme $S$. For $S$ the spectrum of a field, this follows from 0BX7 in the Stacks Project, but what about general $S$?

Answer 1

I don't have a description of the class of morphisms you are describing, but let me provide a counterexample in your case of interest that can perhaps limit your search for a general result. Consider the natural inclusion $f: U = (\mathbb{A}^1_\mathbb{C} \setminus 0) \times \mathbb{A}^1_\mathbb{C} \to \mathbb{A}^1_\mathbb{C} \times \mathbb{A}^1_\mathbb{C} = \mathbb{A}^2_\mathbb{C} = V$, and let $g: X = \mathrm{Bl}_0 \mathbb{A}^2_\mathbb{C} \to \mathbb{A}^2_\mathbb{C}$ be the blow-up of the plane at the origin. The vertical maps are the natural inclusion and the identity, respectively. In this setting there is no lift $h$ because each direction in the plane would need a different value of $h(0)$ for the map to even be continuous.