Isbell duality

Isbell conjugacy (a.k.a. Isbell duality or Isbell adjunction) (named after John R. Isbell ) is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986. That is a duality between covariant and contravariant representable presheaves associated with an objects of categories under the Yoneda embedding. Also, says that; "Then the conjugacies are the first step toward expressing the duality between space and quantity fundamental to mathematics".

Yoneda embedding
The (covariant) Yoneda embedding is a covariant functor from a small category $$\mathcal{A}$$ into the category of presheaves $$\left[\mathcal{A}^{op}, \mathcal{V} \right]$$ on $$\mathcal{A}$$, taking $$X \in \mathcal{A}$$ to the contravariant representable functor:

$$Y \; (h^{\bullet}) :\mathcal{A} \rightarrow  \left[\mathcal{A}^{op}, \mathcal{V} \right]$$

$$X \mapsto \mathrm{hom} (-,X).$$

and the co-Yoneda embedding (a.k.a. contravariant Yoneda embedding  or the dual Yoneda embedding )  is a contravariant functor (a covariant functor from the opposite category) from a small category $$\mathcal{A}$$ into the category of co-presheaves $$\left[\mathcal{A}, \mathcal{V} \right]^{op}$$ on $$\mathcal{A}$$, taking $$X \in \mathcal{A}$$ to the covariant representable functor:

$$Z \; ({h_{\bullet}}^{op}): \mathcal{A} \rightarrow \left[\mathcal{A}, \mathcal{V} \right]^{op}$$

$$X \mapsto \mathrm{hom} (X,-).$$

Every functor $$F \colon \mathcal{A}^\mathrm{op}\to \mathcal{V}$$ has an Isbell conjugate $$F^{\ast} \colon \mathcal{A} \to \mathcal{V}$$, given by

$$F^{\ast} (X) = \mathrm{hom} (F, y(X)).$$

In contrast, every functor $$G \colon \mathcal{A} \to \mathcal{V}$$ has an Isbell conjugate $$G^{\ast} \colon \mathcal{A}^\mathrm{op} \to \mathcal{V}$$ given by

$$G^{\ast} (X) = \mathrm{hom} (z(X), G).$$

Isbell duality
Isbell duality is the relationship between Yoneda embedding and co-Yoneda embedding；

Let $$\mathcal{V}$$ be a symmetric monoidal closed category, and let $$\mathcal{A}$$ be a small category enriched in $$\mathcal{V}$$.

The Isbell duality is an adjunction between the categories; $$\left(\mathcal{O} \dashv \mathrm{Spec} \right) \colon \left[\mathcal{A}^{op}, \mathcal{V} \right] {\underset{\mathrm{Spec}}{\overset{\mathcal{O}}{\rightleftarrows}}} \left[\mathcal{A}, \mathcal{V} \right]^{op}$$.

The functors $$\mathcal{O} \dashv \mathrm{Spec}$$ of Isbell duality are such that $$\mathcal{O} \cong \mathrm{Lan_{Y}Z}$$ and $$\mathrm{Spec} \cong \mathrm{Lan_{Z}Y}$$.