I don't find the "arrow crossing" principle very convincing -- it isn't difficult to adjust the figure so that no arrows cross, even though there's no adjoint. And when the preorders are more complicated than simple total orders, I expect there to be cases where arrows must cross even though there is an adjunction.
It works here because the arrow crossing principle is presumably valid on figures drawn by the authors, else the authors would not provide said principle (or said figures).
**EDIT:** Perhaps it might work more generally if one adds the proviso that in-arrows and out-arrows are _sorted_ in where they attach to the points. But I still suspect there will be edge cases.