Let $F, G: \mathcal{J} \to \mathsf{Sp}$ be continuous functors between $\sf{Sp}$-enriched categories, where $\sf{Sp}$ denotes any of the point-set models for spectra (i.e., orthogonal spectra).

Natural transformations between $F$ and $G$ forms a spectrum given by the enriched end $$ {\sf{nat}}(F,G) = \int_{j \in \mathcal{J}} {\sf{Map}}_{\sf{Sp}}(F(j), G(j)). $$

My question is the following.

What is the Spanier-Whitehead dual of the spectrum ${\sf{nat}}(F,G)$?

As a follow-up. We can also consider the tensor product of enriched functors. $$ \mathbb{D}(G) \otimes_{\mathcal{J}} F = \int^{j \in \mathcal{J}} \mathbb{D}(G(j)) \otimes F(j). $$

Is there a relation between the Spanier-Whitehead dual of the spectrum of natural transformations ${\sf{nat}}(F,G)$ and the tensor product of enriched functor?