  We introduce and describe the $2$-category $\mathsf{Grt}_{\text{flat}}$ of
Grothendieck categories and flat morphisms between them. First, we show that
the tensor product of locally presentable linear categories $\boxtimes$
restricts nicely to $\mathsf{Grt}_{\text{flat}}$. 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.
