My question is a subquestion of this question. And a repost from this MSE question.

The settings is the following. Let $X$ be a scheme. Assume that the adjunction in TAG01BH of the stacks project is actually an adjoint equivalence. My goal is to proof that $X$ is affine.

This is a special case of the general question: can we reconstruct a scheme from its category of quasi-coherent sheafs? I am aware of the available reconstruction theorems, but I feel that this could be done with elementary methods. Feel free to critique this feeling.

For starters, quasi-separatedness of $X$ should be established before you can invoke Gabriel-Rosenberg and quasi-compactness before you can use Serre's vanishing theorem on $X$. Any hints on how to go about this?

What I tried thus far is trying to use Serre's vanishing theorem. The problem: how to prove that there exist 'enough' acyclic resolutions inside $Qcoh(X)$ as is the case for $Qcoh(\mbox{Spec } \Gamma(X,O_X))$, since $\widetilde{I}$ is flasque for injective $\Gamma(X,O_X)$-modules $I$.