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Let $\mathcal{C}$ be a closed symmetric monoidal category and let $PSh(\mathcal{C}):=Fun(\mathcal{C}^{op}, Set)$ 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 $PSh(\mathcal{C})_{fd} \simeq PSh(\mathcal{C}_{fd})$? Here $\mathcal{C}_{fd}$ denotes the subcategory of dualizable objects in $\mathcal{C}$.

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