It seems the link to the note is broken. Do you have a different link or another way I could access the paper?

@July I'm not sure when it was first published, but it's been around for a year or two at least at this point (I forget exactly when I discovered it).

@July: An English translation does exist.

I believe (this agrees with Peter Scholze's "Perfectoid Spaces," at least, and a quick glance at Torsion didn't find anything suggesting otherwise) that the morphisms in the category of perfectoid spaces are simply the morphisms of adic spaces between perfectoid spaces.

I can see the script M now (I am now on my phone), so I think it was originally just M, but either way I'm seeing the script M now so I have no complaints!

"Since clearly $\operatorname{Gal}(M/L)\cong\operatorname{Gal}(M/L)$..." Your identification $M = M\{\{T\}\}$ is confusing. I think you'd help your question by using $\tilde M$ or $M'$ to represent $M\{\{T\}\}$ instead of $M$, seeing as $M$ already has meaning.

You've given me a good bit to think about; I'll probably come back and accept this later after some thought about the counterexample and the comments, and I might wind up posting another related question once I think about what exactly will be done to fix the issue. Thanks Qiaochu!

@QiaochuYuan: This is in analogy with showing that $Lex(\mathcal A,\mathsf{Ab})$ is an abelian category in the proof of the Freyd-Mitchell embedding theorem, so I really do want $Lex(\mathcal B,\mathsf{Set}_*)$ (or at least, the proofs/references I have for F-M use $Lex(\mathcal A,\mathsf{Ab})$ rather than $Lex(\mathcal A^{op},\mathsf{Ab})$).