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:

${\cal O}(\hom(-,x)\amalg \hom(-,y)) = \hom(x,-)\times \hom(y,-)$;

$\text{Spec}$ 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 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.