The notion of a "compact category" was introduced by Isbell$\color{red}{^{1,2}}$. A locally small category $\mathcal{C}$ is called *compact* when every functor $\mathcal{C} \to \mathcal{D}$ into any category $\mathcal{D}$ which preserves all (possibly large!) colimits is a left adjoint. Equivalently, every presheaf $\mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ preserving all (possibly large!) limits is representable.

Personally, I find large limits and colimits a bit awkward (but maybe you can convince me why they are important?), and in my research I need a property which is perhaps stronger: Let's call a category $\mathcal{C}$ *strongly compact* when every functor $\mathcal{C} \to \mathcal{D}$ which preserves small colimits is a left adjoint. This is stronger because we deal with a potentially larger class of functors. Equivalently, every presheaf on $\mathcal{C}$ preserving small limits is representable.

I have done quite a bit of literature research, but did not find this property elsewhere. But it is very natural, in particular in the context of the special adjoint functor theorem, which says that every wellcopowered cocomplete category with a generator is strongly compact. So I suspect that this notion appears somewhere? If not, what name do *you* suggest?




$\color{red}{^1}$ J. R. Isbell, Small subcategories and completeness, Mathematical systems theory 2.1 (1968): 27-50

$\color{red}{^2}$ R. Börger, W. Tholen, M. B. Wischnewsky, H. Wolff, Compact and hypercomplete categories, Journal of
Pure and Applied Algebra 21.2 (1981): 129-144