In 2-category theory we often encounter notions with noninvertible coherence 2-cells, in which case there is a choice about which way the 2-cell should point. Generally, one of these directions is called "lax" and the other "colax" (or "oplax"). We have to make at least one arbitrary choice for which of the two to call "lax" and which to call "colax", but once we've made that choice in one place, the choices everywhere else are determined. One general place to phrase the choice is as defining lax and colax morphisms of algebras for a 2-monad; everything else can be seen as either an instance of that or a generalization of that, and that tells you which direction is lax and which is colax (at least if you want to be consistent, which not everyone does).
At least, that's what I used to think. But today I started thinking about lax dinatural transformations, which should be like ordinary dinatural transformations but with a 2-cell inhabiting the dinaturality hexagon. There are two directions that that 2-cell can go in, but I can't figure out a way to relate those directions to any other situation that has a lax and a colax version.
Since dinatural transformations don't even compose in general, there's no clear way to see them as morphisms of algebras for some monad. The obvious thing to do would be to specialize a dinatural transformation to a natural one by making the two functors either both totally covariant or both totally contravariant, and then compare to the usual terminology in that special case. But there are two ways to do this (covariant or contravariant), and the same direction of 2-cell in a dinaturality hexagon specializes to a lax natural transformation in one case and a colax one in the other case. I suppose we could choose to name it according to the covariant direction because "covariant functors are more basic than contravariant ones", but that feels more arbitrary than I'd like.
You can also specialize a dinatural transformation to an extranatural transformation. But I've looked at the references cited here about lax extranatural transformations, and as far as I can see none of them gives any reason for choosing which direction is lax and which is colax.
It is, of course, true that if you change the domain category to its opposite, permuting the domain to regard a functor $F:C^{\rm op}\times C\to D$ as a functor $F:(C^{\rm op})^{\rm op}\times C^{\rm op}\to D$, then the direction of the 2-cell in a dinaturality hexagon switches. This is not the case for ordinary natural transformations, and suggests that maybe there is more arbitrariness in the dinatural and extranatural case than in the ordinary one.
But I still have to ask, in case anyone else can see something I can't: Is there any principled way to decide which direction of a 2-cell in a dinaturality hexagon (or an extranaturality wedge) is "lax" and which is "colax"?