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Stable category of pro-spaces

I am in the situation of having to understand groups $[\mathbb{S},E\wedge X_+]:=\pi_0\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)$ in $\text{Sp}(\text{Pro}(\mathcal{S}))$, t …