Exponentiable Grothendieck categories in flat Algebraic Geometry

We introduce and describe the $2$-category ${\mathsf{G}\mathsf{r}\mathsf{t}}_{\mathrm{♭}}$ of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories $⊠$ restricts nicely to ${\mathsf{G}\mathsf{r}\mathsf{t}}_{\mathrm{♭}}$. Then, we characterize exponentiable objects with respect to $⊠$: 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 $\mathsf{Q}\mathsf{c}\mathsf{o}\mathsf{h}(X)$ is exponentiable. Finally, we provide a family of examples and concrete computations of exponentials.