This question has been asked by Teimuraz Pirashvili many years ago. I forgot about it after a while and remembered only now by accident. He probably knows the answer by now, but I still don't.

In the category of modules over a ring $R$, the module $R$ is a projective generator. This property does not determine it uniquely, even up to isomorphism, but it does when $R$ is commutative: it can be reconstructed as the ring of endotransformations of the identity functor. (For noncommutative $R$, it is still true that the category is equivalent to the category of modules over the endomorphism ring of any of its projective generators that may exist.)

Now what can be said about the structure sheaf $\mathcal O_X$ of a scheme $X$? Can it be detected inside the category of $\mathcal O_X$-modules without using any additional structure? Of course it is the unit of the monoidal structure, but can it be also characterized as an object in a plain category, without invoking any additional structures? Can in fact the category of $\mathcal O_X$-modules possess other non-isomorphic monoidal structures? Can it be equivalent to the category of $\mathcal O_Y$-modules for some other scheme?

Note that $\mathcal O_X$ is not even a generator - the subcategory generated by it is the category of quasicoherent sheaves. Still also in this category $\mathcal O_X$ is not projective in general. So the same question arises - is it some particular kind of generator?

There are of course several reconstruction theorems but I still cannot figure out which (if any) of them provide answers to these questions.