  Let $k$ be a field. We show that locally presentable, $k$-linear categories
$\mathcal{C}$ dualizable in the sense that the identity functor can be
recovered as $\coprod_i x_i\otimes f_i$ for objects $x_i\in \mathcal{C}$ and
left adjoints $f_i$ from $\mathcal{C}$ to $\mathrm{Vect}_k$ are products of
copies of $\mathrm{Vect}_k$. This partially confirms a conjecture by
Brandenburg, the author and T. Johnson-Freyd. Motivated by this, we also
characterize the Grothendieck categories containing an object $x$ with the
property that every object is a copower of $x$: they are precisely the
categories of non-singular injective right modules over simple, regular, right
self-injective rings of type I or III.
