# Dualizable presheaves with respect to Day convolution

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 $$\mathcal{C}$$.