Interesting example of more than one adjunction between a pair of adjoint functors?

Question

Suppose $ F : \mathcal{C} \to \mathcal{D} $, and $G : \mathcal{D} \to \mathcal{C} $ is a pair of adjoint functors with an adjunction $\alpha : Hom (FX,Y) \simeq Hom(X,GY)$.
Now in principle there could be more than one adjunction between the two functors $F,G$. Are there any common interesting examples where this situation arises ?

Answer 1

The morphism is unique up to an isomorphism; hence the answer to your question is yes and no.

Since there is a unique morphism that can "stand in" for any other, the other isomorphic copies (equivalence classes if you will) are irrelevant; the canonical copy carries all the information you need, hence the unique designation, hence a no answer.

But the fact remains that by definition, there are still isomorphic adjunctions and thus there are in fact other adjunctions. Further, by the Univalent Foundation and it's "equality is isomorphic to isomorphism" theorem, you are allowed to work within these copies instead of just with the equivalence class of copies; with a particular adjunction instead of just with it's categorically canonical version. Hence, a yes answer