Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by $$\{P\setminus{\downarrow x} : x\in P\} \cup \{P\setminus{\uparrow x} : x\in P\},$$ where $\downarrow x = \{y\in P: y\leq x\}$ and $\uparrow x = \{y\in P: y\geq x\}$.

It turns out that every order-preserving function is continuous with respect to the interval topology. So we get a functor ${\bf F}$ from ${\bf Poset}$, the category of partially ordered sets with order-preserving maps, to ${\bf Top}$, the category of topological spaces with continuous functions.

Does ${\bf F}$ have a left adjoint? Does it have a right adjoint?