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## derived classes – Duality between $D^b(mathbb{Z})$ and $D(mathrm{Solid})^omega$

My query is about Corollary 6.1(ii) in Lectures on Condensed Mathematics by Clausen & Scholze (web page 41). Here is the pretense:

The derived class $D(mathrm{Solid})$ is compactly generated, and the complete subcategory $D(mathrm{Solid})^{omega}$ of compact objects consists of the bounded complexes all of whose phrases are of the figure $Pi_{I} mathbb{Z}$. The class $D(mathrm{Solid})^{omega}$ is contravariantly equal to the class $D^{b}(mathbb{Z})$ by way of the functor

$$

C mapsto R underline{operatorname{Hom}}(C, mathbb{Z}) : D^{b}(mathbb{Z})^{mathrm{op}} rightarrow D(mathrm{Solid})

$$

I’m not fairly judgement the way to construe this inner hom. I understand how to perceive $Runderline{operatorname{Hom}}(C, mathbb{Z})$ as an objective in $D^b(mathbb{Z})$, however I’m anticipating to get an objective in $D(mathrm{Solid})$. One route to get from $D^b(mathbb{Z})$ to $D(mathrm{Solid})$ is to first strike to $D(mathrm{Cond}(Ab))$, then apply the derived solidification functor $(-)^{Lblacksquare}$. So ought to I cerebrate of this as

$$C mapsto left(underline{R underline{vphantom{_}operatorname{Hom}}(C, mathbb{Z})}privilege)^{Lblacksquare}?$$

Alternatively, is there some intuition $underline{R underline{vphantom{_}operatorname{Hom}}(C, mathbb{Z})}$ is stable? Or am I fully off abject?

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