It is well known that under the Isbell duality $\text{Spec}\dashv {\cal O} : {\cal V}^{A^°} \leftrightarrows \big({\cal V}^A\big)^°$ representable functors are self-dual, i.e. fixed by the unit and counit of the adjunction: $$\text{Spec}({\cal O}(\hom(-,x)))\cong \hom(-,x)\qquad\qquad {\cal O}(\text{Spec}(\hom(y,-)))\cong \hom(y,-)$$


No mention is made about representables being the *unique* self-dual objects. Is it true?


If not, how to characterize a bigger class of Isbell self-dual objects? My initial guess was that at least a class of colimits of representables are Isbell-self-dual. They are not, even for coproducts, if this computation is correct: 


1. ${\cal O}(\hom(-,x)\amalg \hom(-,y)) = \hom(x,-)\times \hom(y,-)$;
2. $\text{Spec}$ of this evaluated in $a$ now equals
$$
Nat(\hom(-,x)\times \hom(y,-), \hom(a,-))
$$
that doesn't seem related to the initial presheaf. If you assume that $A$ has coproducts, in fact, it is true that this is $\hom(a,x\amalg y)$, so that $\text{Spec}({\cal O}(h\_x\amalg h\_y)) \cong h\_{x\amalg y}$, different from $h\_y\amalg h\_y$ in general.


This question has already been [asked](https://math.stackexchange.com/questions/1953264/examples-of-isbell-self-dual-objects) on math.SE, and it was me that pointed the OP to another MO thread discussing the topic, but I'm seeking some explicit answer now.



Characterizing Isbell self-dual objects
A quick answer for now, which I might add more to later. Recall that for every adjunction $F \dashv G: C \to D$ there is the notion of "fixed point" of the adjunction which has two faces: either it is an object $c$ of $C$ for which the counit $\epsilon\_c: FGc \to c$ is an isomorphism, or it is an object $d$ of $D$ for which the unit $\eta\_d: d \to GFd$. The adjunction $F \dashv G$ then induces an adjoint equivalence between the full subcategories $\text{Fix}\_{FG}(C)$ and $\text{Fix}\_{GF}(D)$, and thus we identify these categories. In the case of the Isbell conjugation adjunction for a small category $X$, the category of fixed points is often called the *Isbell envelope* or *Isbell completion*; let's denote it $I(C)$. 


In the special case where $C$ is a preorder, i.e., a $\mathbf{2}$-enriched category, $I(C)$ is a preorder which goes by a more famous name: the Dedekind-MacNeille completion. You can find some discussion of this in the MO-answer [here](https://mathoverflow.net/a/59295/2926). In this case $I(C)$ *is* both complete and cocomplete (i.e. admits small limits and colimits). 


In other cases, the Isbell completion need not be $\mathcal{V}$-complete/cocomplete, as you correctly surmise (see comments below the MO-answer I just mentioned), but it can be quite a bit larger (and certainly larger than the Cauchy completion). It's maybe best to point to some examples. For the case where $\mathcal{V}$ is the monoidal closed category $([0, \infty), \geq, +)$, i.e., where $\mathcal{V}$-categories are metric spaces in the sense of Lawvere, Simon Willerton has an interesting article which connects the Isbell completion to the tight span of metric spaces; you can find a Café discussion [here](https://golem.ph.utexas.edu/category/2013/01/tight_spans_isbell_completions.html) where some examples are computed. 


Isbell completions also figure in "[comparative smootheology](https://arxiv.org/pdf/0802.2225v2.pdf)", but I'd need to study and think more before commenting on that. You can also find more references to the Isbell completion in the nLab; as I said, I may come back and add more. 


