My question is a subquestion of this question. I do not want to use full reconstruction theorems.
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. As is the case for every affine scheme. Is $X$ affine? Or, which is the same, is the canonical map $X\rightarrow \mbox{Spec } O_X(X)$ an iso?
What I tried thus far is proving all hypothesis of Serre's vanishing theorem and invoking it. The problem: how to prove that there exist 'enough' acyclic resolutions inside $Qcoh(X)$ as is the case for $Qcoh(\mbox{Spec } O_X(X))$ since $\widetilde{I}$ is flasque for injective modules $I$. Any hints?
Any hints in general are also welcome.
