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This question was posted on MSE and got very little attention, so I'm also posting it here.

Let $\mathcal{C}$ be a closed symmetric monoidal category and let $PSh(\mathcal{C}):=Fun(\mathcal{C}^{op}, \mathbf{Set})$ be its category of presheaves regarded as a closed symmetric monoidal category via Day convolution of presheaves.

Is there a nice description of the dualizable objects of $PSh(\mathcal{C})$ in terms of the dualizable objects of $\mathcal{C}$? For example, could it be that the dualizable presheaves $PSh(\mathcal{C})_{fd}$ consists of objects given as filtered (co)limits of dualizable objects in $\mathcal{C}$?

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Lemma: In a closed symmetric monoidal category where the unit object $1$ is tiny (meaning $\text{Hom}(1, -)$ preserves colimits), every dualizable object is tiny.

Proof. If $x$ is dualizable, then $\text{Hom}(x, -) \cong \text{Hom}(1, x^{\ast} \otimes (-))$. Since the monoidal structure is assumed to be closed, $x^{\ast} \otimes (-)$ preserves colimits, and by assumption so does $\text{Hom}(1, -)$. $\Box$

The unit object in $\text{Psh}(C)$ is the presheaf represented by the unit object; since colimits in presheaf categories are computed pointwise, every representable presheaf is tiny, so the lemma applies and we conclude that every dualizable object in $\text{Psh}(C)$ is tiny. But it's standard that the tiny objects in $\text{Psh}(C)$ are precisely the retracts of the representable presheaves. Among these, the dualizable objects include the presheaves represented by dualizable objects in $C$, as well as (I think?) their retracts. I don't know if it's possible for a nontrivial retract of a presheaf represented by a non-dualizable object to be dualizable.

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    $\begingroup$ There are plenty of dualizable retracts of non-dualizable objects (finite dimensional subvector spaces of infinite-dimensional ones). However, the idempotent completion of $C$ has a natural monoidal structure, and your argument shows that the dualizable objects of $Ph(C)$ are those of the idempotent completion of $C$. And one cannot do better: it is possible that C has no non-trivial dualizable objects while its idempotent completion has many (e.g. C=infinite dimensional vector spaces with countable basis). $\endgroup$
    – D.-C. Cisinski
    Commented Oct 11, 2018 at 13:36

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