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

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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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