Exponentiable Grothendieck categories in flat Algebraic Geometry

We introduce and describe the $2$-category $mathsf{Grt}_{flat}$ of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories $boxtimes$ restricts nicely to $mathsf{Grt}_{flat}$. Then, we characterize exponentiable objects with respect to $boxtimes$: these are continuous Grothendieck categories. In particular, locally finitely presentable Grothendieck categories are exponentiable. Consequently, we have that, for a quasi-compact quasi-separated scheme $X$, the category of quasi-coherent sheaves $mathsf{Qcoh}(X)$ is exponentiable. Finally, we provide a family of examples and concrete computations of exponentials.