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?