Let $\mathcal{C}$ be a discrete index category and $\mathcal{S}$ the category of (e.g.) orthogonal spectra. We consider the homotopy category $\mathrm{Ho}(\mathrm{Fun}(\mathcal{C,S}))$ of diagram spectra indexed by $\mathcal{C}$.
An object $T$ is as usual called compact if for any small family $(X_i)_{i\in I}$, the natural map $$\bigoplus [T,X_i] \rightarrow [T, \coprod X_i]$$ is an isomorphism.
A finite object is an object of the form $\Sigma^N\Sigma^{\infty}A$ where $A$ is a finite $\mathcal{C}$-CW-complex, up to isomorphism in the homotopy category. Every finite object is compact.
Question: Is every compact object finite?
