From what I could see by googling around, I bet you're right. I'm afraid this could mean that the proof is very indirect... By the way what does finely mean?

I actually meant the category has finitely many morphisms. It's the kind of homotopy limits that commute with sequential homotopy colimits, maybe we can ask for a very small category to be safe.

Oh,homotopy colimits are homotopy invariant even in spectra without cofibrant replacements

That's precisely what I was asking. Is it known not to be homotopy invariant for spectra in spaces? I see how it is not for spectra in simplicial sets. Funny things can happen, e.g. homotopy colimits are homotopy invariant in spaces even without cofibrant replacements

It makes sense even without having a model structure

I'm using the Bousfield-Kan formula to define homotopy limits. Since the cotensored structure of spectra over simplicial sets is levelwise, the homotopy limits turn out to be levelwise (and well defined). The proof uses only that stable equivalences are $\pi_\ast$-equivalences, and that directed homotopy colimits and finite homotopy limits commute in spaces.

@Ricardo: You're right, The functor $F$ I want to consider is not the standard inclusion. The small category should contain $m$ as final object. I am going to edit the question.

Yeah it is not a great example. The map is null homotopic means that it is an iso in $\pi_k$ for $k$ below the minimum of the connectivities of $\Omega^nY$ and $\Omega^mY$. This is the same as the connectivity of $Y$ minus $m=\dim S^m/S^n$ for $n<m$.

Yeah, I know his work! I was thinking precisely of a variation of his argument. The fixed point he uses are defined as a limit, and this gives the wrong notion for G-spectra (they are just the levelwise fixed points). But the geometric fixed points functors give a family of adjunctions that might define a model structure. For it to be the right one I need the question I asked to be true :)

If it was true, it might give a chance to define the model structure on orthogonal G-spectra as a transport construction from the model structure on orthogonal spectra. I think this could be quite convenient.