(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?

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Let V and W be cosmoi (bicomplete closed symmetric monoidal categories) and let F:V→W and G:W→V be lax monoidal functors with a monoidal adjunction F⊣G (hence F is strong). Then F and G induce 2-functors F: V-Cat → W-Cat and G: W-Cat → V-Cat and lax functors F: V-Prof → W-Prof and G: W-Prof → V-Prof between the corresponding bicategories of enriched categories and profunctors (= "modules"). The functor F simply applies F to all the hom-objects of a V-category or V-profunctor, and likewise for G.

Now for any V-category X and W-category Y, we have a functor φX,Y: W-Prof(FX,Y) → V-Prof(X,GY), defined for a W-profunctor M:FX⊗Yº → W by first applying G to get a V-profunctor M:GFX ⊗ GYº → V, and then composing with the unit η: X → GFX of the adjunction between V-Cat and W-Cat. This functor has a left adjoint defined for a V-profunctor N:X⊗GYº → V by first applying F to get a W-profunctor N:FX⊗FGYº → W, and then left Kan extending along the counit ε:FGYº → Yº of the adjunction between V-Cat and W-Cat. Thus, the functors F and G between V-Prof and W-Prof are "lax-2-adjoint".

I would personally prefer to phrase this as a more ordinary sort of adjunction between equipments; in fact one can show that any adjunction between equipments gives rise to such a "lax-2-adjunction" between their categories of proarrows. But it is a "naturally occurring" example of a "lax-2-adjunction." (I put "lax-2-adjunction" in quotes because I regard the terminology as not settled. Note that "lax-2-adjunctions" are also called "local adjunctions," e.g. in Betti and Power, "On local adjointness of distributive bicategories.")

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SEE: Soft adjunction between 2-categories (John L. MacDonald and Arthur Stone) JPAA 60 (1989) p. 155-203

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

Anyway this paper is very theoric, but give some equivalents formulation of generalizated adjunctions between 2-categories.

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