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,

- 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.
- Where exactly does the right adjoint described in Proposition 10.5 come from?

Many thanks!

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