Suppose we have the category $\mathrm{sSet}$ of simplicial sets and $\mathrm{sSpectra}$ of simplicial spectra. Then we have the functor $\Sigma^{\infty} \colon \mathrm{sSet} \to \mathrm{sSpectra}$, $X \mapsto \Sigma^{\infty} X$, where $(\Sigma^{\infty} X)_n = \Sigma^n X = S^1 \wedge X$. We can then go to the pro-categories $\mathrm{Pro}(\mathrm{sSet})$ and $\mathrm{Pro}(\mathrm{sSpectra})$ of Pro-objects and get the functor $\Sigma{\infty} \colon \mathrm{Pro}(\mathrm{sSet}) \to \mathrm{Pro}(\mathrm{sSpectra})$ by applying $\Sigma{\infty}$ to each component of a pro-object.
We have now cohomology of pro-objects, via $H^*(X;M) = \mathrm{colim}_i H^*(X_i; M)$ for a pro-object $X$ and an abelian group $M$ (regardless if $X$ is a pro-simplicial set or a pro-spectrum). A $\mathbb{F}_2$ cohomologycal weak equivalence is a map $X \to Y$ which induce an isomorphism $H^*(Y; \mathbb{F}_2) \to H^*(X; \mathbb{F}_2)$. I want to prove that $\Sigma^{\infty}$ preserves these $\mathbb{F}_2$ cohomological weak equivalences. Does anybody know how to do that?