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 (and hence preserves all colimits). This is stronger because we deal with a a priori 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? An alternative would be "realized-sketchable" since I can show that a category is strongly compact if and only if it is the category of models of a (possibly large) realized limit sketch for which every model is small.

*Edit. In my paper I am using the name strong compact.

$\color{red}{^1}$ 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