Choiceless L\"owenheim-Skolem property and uniform definability of grounds

In this paper, without the axiom of choice, we show that if a certain downward L"owenheim-Skolem property holds then all grounds are uniformly definable. We also prove that the axiom of choice is forceable if and only if the universe is a small extension of some transitive model of $\mathsf{Z}\mathsf{F}\mathsf{C}$.