Assuming $\mathcal C$ and $\mathcal D$ are locally small and small-cocomplete, a functor $F \colon \mathcal C \to \mathcal D$ has a left adjoint if and only if it is small-cocontinuous and satisfies the solution set condition. Therefore, a category $\mathcal C$ is strongly compact in your sense if and only if every small-cocontinuous functor with domain $\mathcal C$ satisfies the solution set condition. According to the terminology of Di Liberti–Loregian in [Accessibility and presentability in 2-categories](https://arxiv.org/abs/1804.08710), these are categories that are *petit* with respect to small presheaves (see Remark A.2 ibid.). A locally small category $\mathcal C$ is *petit with respect to small presheaves* if, for all functors $F \colon \mathcal C \to \mathcal D$, the induced functor between categories of small presheaves $\widehat F \colon \widehat C \to \widehat  D$ is left adjoint (explicitly, $\widehat F = \mathrm{Lan}_{y_{\mathcal C}} (y_{\mathcal C} \circ F)$). Various equivalent definitions may be found in Definition 22 of Walker's [Distributive laws via admissibility](https://arxiv.org/abs/1706.09575).

There is one caveat, which is that the correspondence between petitness with respect to small presheaves and the solution set condition is only conjectured in that paper, not proven. I believe the authors have since proven this result, though it is as yet unpublished. (I admit that, without the proof, this answer is a little unsatisfactory!)