  This is our second scope of the consideration on the corresponding
topologization and the corresponding functional analytification. We will focus
on the corresponding functorial and motivic constructions in our current
consideration. We consider topological motivic derived $I$-adic and derived
$(p,I)$-adic cohomologies through derived de Rham complexes of Bhatt, Guo,
Illusie, Morrow, Scholze, Frobenius sheaves over Robba rings of Kedlaya-Liu in
certain derived $I$-adic and derived $(p,I)$-adic geometric context as what we
defined for Bambozzi-Ben-Bassat-Kremnizer $\infty$-prestacks in our previous
work in this series. The foundation we will work on will be based on the work
of Bambozzi-Ben-Bassat-Kremnizer, Ben-Bassat-Mukherjee, Clausen-Scholze and
Kelly-Kremnizer-Mukherjee, in order to promote the construction to even more
general homotopical and $\infty$-categorical contexts. This gives us the chance
to construct the functional analytic derived prismatic cohomology and derived
preperfectoidizations, as well as the functional analytic derived logarithmic
prismatic cohomology and derived logarithmic preperfectoidizations after
Bhatt-Scholze and Koshikawa, in the framework of Bambozzi-Ben-Bassat-Kremnizer,
Ben-Bassat-Mukherjee, Clausen-Scholze and Kelly-Kremnizer-Mukherjee.
