Provided a small category $G$, the Isbell duality establishes an adjunction:
$$\text{Set}^{G^{\text{op}}} \leftrightarrows (\text{Set}^{G})^{\text{op}}. $$
Both maps can be computed as Kan extensions. Call $y : G \to \text{Set}^{G^{\text{op}}} $ and $y^{t}: G \to (\text{Set}^{G})^{\text{op}} $ the Yoneda embeddings, then one can obtain the two adjoints as $\text{Lan}_y y^t$ and $\text{Lan}_{y^t} y.$
I am interested in a special computation of these maps. Choose $G$ to be the terminal category, so that we are looking at the adjunction:
$$\text{Set} \leftrightarrows \text{Set}^{\text{op}} .$$
Is there an explicit presentation of these two adjoint functors?
Every set is a coproduct of points, and the left adjoint sends the point to itself, so it sends any set to a product of points, that is, it's constant. Similarly, or by adjunction, the right adjoint is constant. There's an error in your treatment: the opposite category of sets is the free completion of a point under limits, and so the right adjoint must be defined by right Kan extension along $y^t$.