  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.
