**JL1:** Let \$$U\$$ be an upset, and let \$$x \in \operatorname{\uparrow}(U)\$$. Then there is some \$$y \in U\$$ such that \$$y \le x\$$. Since \$$U\$$ is an upset, this means that \$$x \in U\$$. Therefore, \$$U = \operatorname{\uparrow}(U)\$$. So every upset \$$U\$$ is of the form \$$\operatorname{\uparrow}(S)\$$ for some \$$S\$$; at the very least, take \$$S\$$ to be \$$U\$$ itself, forgetting its order structure.

I think there’s an adjoint relationship happening here too, between \$$\operatorname{\uparrow}\$$ and the forgetful function from posets to sets. In fact, \$$\operatorname{\uparrow}\$$ also seems to be a closure operator, which as someone else noted is what monads are for posets.