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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 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, these are categories that are petit with respect to small presheaves (see Remark A.2 ibid.). 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 I believe the authors have since proven this result, though it is as yet unpublished.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!)