(There are several dialects of 2-categorical language. Mine is the one where everything is weak by default. You might need to insert the word "weak" or "pseudo" (or even "strong", if your default is lax) here and there to translate this to your own dialect.)

The usual notion of an adjunction between two 2-categories C and D is a pair of 2-functors F : C → D and G : D → C together with a suitably natural equivalence between the categories HomD(FX, Y) and HomC(X, GY). One could ask for something weaker—rather than an equivalence, just a natural functor φX,Y : HomD(FX, Y) → HomC(X, GY) which itself has a right adjoint. (We could instead ask for φX,Y to have a left adjoint; this gives a different notion for any particular C and D, but we can interchange the two notions by reversing all the 2-morphisms in C and D, so we'll just pick this notion arbitrarily.) This is called a "lax 2-adjunction" at the nlab.

A boring example: If D = • is the final 2-category, then an adjunction C → D is a final object of C, while an adjunction up to adjunction is merely an object Z of C such that HomC(X, Z) has an initial object for every X and these initial objects are preserved by precomposition by f : X' → X.

Does anyone know of a more interesting, natural example?

At p. 168 there is a classical example involving the 2-categories $Cat$, and $Adj$ of categories and adjunctions.