TITLE: Dualizable object in the category of locally presentable categories
QUESTION [23 upvotes]: The bicategory of locally presentable categoires, and left adjoint functor between them, is monoidal closed for the Kelly tensor product. My question is what are the dualizable objects for this monoidal structure ?
Just to be clear, I'm talking about the monoidal stucture for which the internal hom $[A,B]$ is given by the category of left adjoint functor $A \rightarrow B$; or equivalently for which  morphismw $A \otimes B \rightarrow C$ are the functors $A \times B \rightarrow C$ that commutes to colimits in each variables.
A typical example of such a dualizable object are the presheaves categories. If $C$ is small then the presheaf category $\widehat{C}$ is dualizable with dual $\widehat{C^{op}}$. The evaluation
$$\widehat{C} \otimes \widehat{C^{op}} \simeq \widehat{C \times C^{op}} \rightarrow Sets $$
Is given by the coend, and the counit:
$$ Sets \rightarrow \widehat{C} \otimes \widehat{C^{op}} \simeq \widehat{C \times C^{op}} $$
Is the unique left adjoint functor sending the singleton to $Hom_C( \_ ,\_ )$. The usual unit/counit relation being essentially the coend formulation of the Yoneda lemma.
I suspect there are some dualizable object not of this form, but this is not totally clear to me, and I wonder if it is possible to give a nice description of the dualizable objects in general...

REPLY [6 votes]: The following is a completement to the answer by Yonatan. I managed to work it out after I read what he had done.
It is basically about proving the converse to the implication he showed in his answer: If I'm correct one can prove that for a locally presentable category $A$, the following are equivalent:
$(1)$ $A$ is dualizable in $Pres^L$
$(2)$ $A$ is a retract of a presheaf category in $Pres^L$
$(3)$ $A$ can be constructed from an ideal $C_0 \subset C$ as described in Yonatan answer.
This being said, I'm not excluding that a better result can be obtained (for example a more canonical presentation of dualizable objects), though the example given by Yonatan seem to suggest that this is already quite close to be optimal.
$(3) \Rightarrow (2) \Rightarrow (1) $ is discussed in Yonatan answer, I'll prove $(1) \Rightarrow (2)$ and $(2) \Rightarrow (3)$ separately.

For $(1) \Rightarrow (2)$. Let $A$ be a dualizable object in $Pres^L$, its dual $A^*$ is isomorphic to the category of left adjoint functors $[A,Set]$. The fact that $A$ is dualizable means that there is a coevaluation map $Set \rightarrow A \otimes A^*$, which is fully described by the image of the singleton, giving a specific object $c \in A \otimes A^*$.
As any object of $A \otimes A^*$, $c$ can be written as a colimits of pure tensor:
$$ c = colim_{i \in I} a_i \otimes \chi_i $$
where $a_i \in A$ and $\chi_i \in A^* = [A,Set]$. The evaluation-coevaluation relation translate into:
$$ colim_{i \in I} \chi_i (u) \times a_i \simeq u $$
functorially in $ u \in A$ (where $\times$ above denote the cotensoring objects by sets, i.e. the coproduct of several copies of the same object).
It means that the $\chi_i$ together form a cocontinuous functor $A \rightarrow Set^I$, And $F \mapsto colim F(i) \times a_i$ form a cocontinuous functor from $Set^I$ to $A$, whose composite $A \rightarrow Set^I \rightarrow A$ is naturally isomorphic to the identity, hence $A$ is a retract of $Set^I$.

I now give a sketch of proof for $(2)  \Rightarrow (3)$:
Assume that one has two morphisms in $Pres^L$, $f:A \rightarrow \widehat{C}$ and $g:\widehat{C} \rightarrow A$ with $g \circ f \simeq_{\Theta} Id$.
Then $P = f \circ g : \widehat{C} \rightarrow \widehat{C}$, is idempotent $P^2 \simeq_{f \theta g} P $, and moreover the isomorphism is an associative multiplication $P^2 \rightarrow P$ (I'm goging to say that $P$ is a non-unital monad), and the category $A$ identifies with the algebras for this multiplication (in the non unital sense), whose structure maps is are isomorphisms $PX \overset{\sim}{\rightarrow} X$.
Using that $P$ commutes to coproduct, this makes $P \coprod Id$ into a cocontinuous (unital) monad on $\widehat{C}$, whose algebras are the $P$-algebra (as a non-unital monad), and hence one has a category $C_P$, whose objects are these of $C$ and morphism from $c$ to $c'$ are morphisms from $c$ to $(Id \coprod P) c'$, i.e. either morphism from $c$ to $c'$, or morphism from $c$ to $P c'$ (composition of the morphism of the second type involving the use of the multiplication maps $P^2  \rightarrow P$
The map of the form $c \rightarrow P c'$ form an ideal in $C_P$, and because of the relation $P^2 \simeq P$ they satisfies the additional condition in Yonatan answer.
Now presheaves over $C_P$ are the same as presheaves over $C$ with an algebra structure for the monad $Id \coprod P$, i.e. algebra for the non-unital monade $P$. One can then check that the additional condition in Yonatan answer (with respect to the ideal mentioned above) single out the $P$-algebra for which the structure map is an isomorphism, and hence it corresponds exactly to the category $A$ we started from.