Condensed Learning Seminar

For a general complete Huber pair $(A,A^+)$, define the analytic ring $(A,A^+)_\blacksquare$. Prove that the associations $\mathrm{Spa}\, (A,A^+)\mapsto \mathcal{D}((A,A^+)_\blacksquare)$, $\mathrm{Spa}\, (A,A^+)\mapsto \mathcal{D}^\omega((A,A^+)_\blacksquare)$, and $\mathrm{Spa}\, (A,A^+)\mapsto \mathcal{D}^\mathrm{nuc}((A,A^+)_\blacksquare)$ satisfy analytic descent. Deduce from that the association $\mathrm{Spa}\, (A,A^+)\mapsto \mathrm{Perf}\, A$ satisfies analytic descent as well.