Silly question, but how does $\mathbb{F}_p^{\Omega^2S^3}$ relate to $H^\ast(\Omega^2 S^3; \mathbb{F}_p)$?

@SimonHenry yeah that makes perfect sense for getting the comonoid structure on $(X,\ast)$, thanks!

I believe I know how to produce a morphism $(X,f)\to (X,f)\otimes (X,\ast)\simeq (X\times X, \mu_Z\circ (f\times \ast)\circ \Delta)$ in the $\infty$-category $C_{/Z}$, but it's not clear to me how to extend it to a fully structured comodule coaction.

Coming back to this I think it's actually kind of straightforward. Given the right sorts of fibrant replacements, that totalization will definitely be the homotopy pullback (in the case of spaces). The question about the BKSS is whether or not it correctly computes the homotopy or homology groups of that space.

The same statement holds for any spectrum $E$ such that $\langle E\rangle < \langle H\mathbb{F}_p\rangle$ (these are the Bousfield classes) I believe. So you have a whole slew of spectra that satisfy the equation $x^2=0$, basically by taking the Brown-Comenetz dual of any connective spectrum with finitely generated homotopy groups.

Doesn't the Brown-Comenetz dual of the sphere, $I$, have the property that $I\wedge I\simeq 0$? So I guess it's integral?

I was just coming back to this (2 years later!) and wondering if you knew how precisely to get that functor Coalg(S)→Pr. I found the relevant functor in for Alg in higher algebra.