Let $X$ be a scheme. Is the category $QCoh(X)$ of quasi-coherent sheaves on $X$ locally presentable? If so, can we say anything about the $\kappa$ for which $QCoh(X)$ is locally $\kappa$-presentable? (e.g. is it always finitely presentable? Or related to the $\kappa$ of [Gabber's result](https://stacks.math.columbia.edu/tag/077K)?)


I'm particularly interested in the case where $X$ is quasi-comact quasi-separated (qcqs).




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In my searching for references, I've come across answers ranging from "we don't know", "when qcqs", to "always", and would appreciate some clarity.



Is Qcoh(X) locally presentable?
Zariski descent tells us that


$$\operatorname{QCoh}(X)=\lim\_{U\subseteq X} \operatorname{QCoh}(U)$$


where $U$ ranges through all open affines and the limit is taken in the $(2,1)$-categorical sense. Since small limits of presentable categories are presentable and $\operatorname{QCoh}(\operatorname{Spec}R)=\operatorname{Mod}\_R$ is presentable, we have that that $\operatorname{QCoh}(X)$ is presentable.


I'm not sure how to get bounds on the accessibility degree in general, though when $X$ is qcqs it is compactly generated (i.e. $\omega$-presentable) by the argument in [this answer](https://mathoverflow.net/a/39952/43054) by Denis-Charles Cisinski.


PS: following recent trends in homotopy theory, I'm dropping the "locally" from "locally presentable'.


