How are called

monotone maps of partially ordered sets such that the image of a lower set is a lower set?

More precisely, I am interested in maps $f:X\rightarrow Y$ of preorders such that (i) $x\leq y$ implies $f(x)\leq f(y)$ (ii) $x\leq f(y)$ implies there is $x'\leq y$ such that $x=f(x')$.

Is there a classification of such maps between finite preorders? Is there a name for them?

The motivation for the question is that these are equivalent to closed maps of finite topological spaces.

Update: The category of finite Esakia spaces is the category of partial orders with maps of this kind; the category of (all) Esakia spaces is dual to the category of Heyting algebras. I found this following Emil Jeřábek comment (thanks!). In modal logics these are known as p-morphisms.

Still, I would like to see terminology by poset people and a reference to classification.
https://en.wikipedia.org/wiki/Esakia_space