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I want to understand the usage of $\infty$-categories in the proof of Proposition 10.5 in the Condensed Mathematics lecture notes available here: https://www.math.uni-bonn.de/people/scholze/Condensed.pdf .

I am struggling with the following sentence in the second paragraph:

It is formal that $F$ has a right adjoint, which is given by taking a collection of objects $M_I \in \mathcal{C}_I$, compatible under localization, to their limit $\varprojlim M_I \in \mathcal{C}$.

I would like to understand where this comes from. That is,

  1. Why are the objects of $\varprojlim_{I}\mathcal{C}_I$ given by collections of objects $M_{I}\in\mathcal{C}_I$, compatible under localization? I am aware of Corollary 3.3.3.2 in Lurie's Higher Topos Theory, which gives an explicit description of limits in the $\infty$-category of $\infty$-categories. However, I do not understand how this relates to the Proposition 10.5 in the Condensed Maths lecture notes, nor how to use Lurie's result in practise.
  2. Where exactly does the right adjoint described in Proposition 10.5 come from?

Many thanks!

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    $\begingroup$ The general limit is covered in Lurie's Higher Algebra 5.2.2.36, but it is easier to see an example how it is used: Lemma D.3.5.5 in Lurie's Spectral Algebraic Geometry. The case in Condensed.pdf is particularly simple because all maps are localizations (of categories). $\endgroup$
    – Z. M
    Commented Apr 13, 2023 at 8:04
  • $\begingroup$ @Z. M many thanks for the references! Could you please explain how the (b) in the proof of Lemma D.3.5.5 in SAG is related to HA 5.2.2.36? $\endgroup$
    – user141099
    Commented Apr 13, 2023 at 19:51
  • $\begingroup$ The (b) in the proof of [SAG, D.3.5.5] is checking the condition (b) in [HA, 5.2.2.36]. Checking that $\phi$ is an equivalence for every $i$ is showing that they form coCartesian edges. The construction per se is a relative limit (note that this coCartesian fibration is in fact also a Cartesian fibration): some result in HTT reduces relative colimits to colimits on fibers, but I don't find it now. $\endgroup$
    – Z. M
    Commented Apr 13, 2023 at 20:40
  • $\begingroup$ @Z.M Many thanks! $\endgroup$
    – user141099
    Commented Apr 18, 2023 at 7:30
  • $\begingroup$ @Z.M So I thought about [HA, 5.2.2.36] for a while, and I am still not precisely sure about the details in the application to [SAG, D.3.5.5]. Also, I do not think this answers my original questions. Maybe could you please clarify again to me how to answer my two questions 1 and 2 from above? To be really precise here, I want to understand how to realise objects in the limit $\infty$-category $\varprojlim_{I}\mathcal{C}_{I}$ as diagrams $S\rightarrow\cal{C}$, in light of HTT 3.3.3.2. $\endgroup$
    – user141099
    Commented May 16, 2023 at 13:49

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