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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 $\text{Sp}(\text{Pro}(\mathcal{S}))$, the stabilization of the $\infty$-category of pro-spaces. Here, $E$ is an object of $\text{Sp}(\text{Pro}(\mathcal{S}))$, and $X$ is a pro-space. Suppose $X=\{X_{\alpha}\}$ is a projective system of spaces. It is true that $\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)=\text{holim}_{\alpha} \text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_{\alpha+})$. If the groups $[\mathbb{S},E\wedge X_{\alpha+}]$ are all finite, then I suspect that we would then have $[\mathbb{S},E\wedge X_+]=\text{lim}_{\alpha} [\mathbb{S},E\wedge X_{\alpha+}]$, but is there a Mittag-Lefler type of argument for when the indexing set is not sequential?

I edited the question in response to Tyler Lawson's comment.

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    $\begingroup$ You should get an identity of mapping spaces, but $\pi_0$ doesn't commute with limits (e.g. the Milnor $lim^1$-sequence for pro-systems indexed by the natural numbers). $\endgroup$
    – Tyler Lawson
    Commented Feb 17, 2017 at 0:27
  • $\begingroup$ I suppose I can try to use a holim spectral sequence for the editted question. $\endgroup$
    – user104997
    Commented Feb 17, 2017 at 1:37

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