I am trying to understand the proof of lemma 2.2.18 in Mann thesis [1]. its statement is surprising for me because it talks about the general rings which are not necessarily characteristic $p$ and out of nowhere Frobenius appear in the statement. the statement is this:

Let $A$ be a ring and $C\subset D_0(A)$ is a subcategory which is stable under cofibers and $L:D_0(A)\to C$ is left adjoint to the forgetful functor and satisfy these conditions:1)if $L(Q)=0$ then $L(Q\otimes M)=0$ for every $M$. 2)if $L(Q)=0$ then for every prime $m$ and every integer $m$, $L\phi_{p^m}^{*}M=0$. then the map from $Ring_C\to Ring_A$ adimits a left adjoint(Ring_C is the category of $A$ algebras whose underlying "module" lies in $C$).

The proof is based on the fact that if $L(Q)=0$ then $L(Sym(Q))=0$. for the proof of this fact, it gives a refrence to analtyc geometry of scholze and clausen where they only give a sketch of the proof and I can't understand it. Can someone explain the proof of this fact in more detail or give a reference?


  [1]: https://arxiv.org/pdf/2206.02022.pdf