Proposition 5.5 in $\mathbf{A}^1$-homotopy theory establishes the Brown's representability for the stable homotopy category $\mathcal{SH}_T(S)$, over a Noetherian scheme $S$, for a space $T$ of finite type over $S$.

Given a closed symmetric monoidal model category $(\mathcal{C},\mathcal{M})$, and an object $T\in \mathcal{C}$. Is there a reference proving when the stable homotopy category of $T$-symmetric spectra in $\mathcal{C}$, as in Spectra and symmetric spectra in general model categories, satisfies the Brown's representability. For instance, is the Brown's representability satisfied when $T$ is compact (for some notion of compactness) and when $\mathcal{M}$ is 'well-behaved' (e.g. cofibrantly generated, simplicial, proper,...)?

I recall reading such a statement, but I do not recall the source, or whether it was proven. So, I would be grateful if you could provide a reference.

cogroupgenerators [and so already satisfies Brown representability], and (2) if T is a cogroup object. $\endgroup$ – Dylan Wilson Feb 18 '17 at 22:15