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Assuming $\mathcal C$ and $\mathcal D$ are locally small and $\mathcal C$ is small-cocomplete, a functor $F \colon \mathcal C \to \mathcal D$ is a left adjoint if and only if it is small-cocontinuous and satisfies the solution set condition. Therefore, a small-cocomplete 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. These categories were studied in Ulmer's The adjoint functor theorem and the Yoneda embedding (see Theorem 13, for instance), but were not given a name.

The relationship to compactness in the sense of Isbell, and totality, is described in Theorem 8 (where functors preserving all large colimits are called supercocontinuous). In particular, compactness is equivalent to totality, so your notion of strong compactness is a form of "small totality".

(Thanks to Ivan Di Liberti for pointing out a mistake in the original answer.)

Assuming $\mathcal C$ and $\mathcal D$ are locally small and $\mathcal C$ is small-cocomplete, a functor $F \colon \mathcal C \to \mathcal D$ is a left adjoint if and only if it is small-cocontinuous and satisfies the solution set condition. Therefore, a small-cocomplete 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. These categories were studied in Ulmer's The adjoint functor theorem and the Yoneda embedding (see Theorem 13, for instance), but were not given a name. Ulmer did not study the variant where $\mathcal C$ is not required to be small-cocomplete.

The relationship to compactness in the sense of Isbell is described in Theorem 8 (where functors preserving all large colimits are called supercocontinuous).

(Thanks to Ivan Di Liberti for pointing out a mistake in the original answer.)

Assuming $\mathcal C$ and $\mathcal D$ are locally small and small-cocomplete, a functor $F \colon \mathcal C \to \mathcal D$ is 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. These categories were studied in Ulmer's The adjoint functor theorem and the Yoneda embedding (see Theorem 13, for instance), but were not given a name.

The relationship to compactness in the sense of Isbell, and totality, is described in Theorem 8 (where functors preserving all large colimits are called supercocontinuous). In particular, compactness is equivalent to totality, so your notion of strong compactness is a form of "small totality".

(Thanks to Ivan Di Liberti for pointing out a mistake in the original answer.)

Assuming $\mathcal C$ and $\mathcal D$ are locally small and $\mathcal C$ is small-cocomplete, a functor $F \colon \mathcal C \to \mathcal D$ is a left adjoint if and only if it is small-cocontinuous and satisfies the solution set condition. Therefore, a small-cocomplete 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. These categories were studied in Ulmer's The adjoint functor theorem and the Yoneda embedding (see Theorem 13, for instance), but were not given a name.

The relationship to compactness in the sense of Isbell, and totality, is described in Theorem 8 (where functors preserving all large colimits are called supercocontinuous). In particular, compactness is equivalent to totality, so your notion of strong compactness is a form of "small totality".

(Thanks to Ivan Di Liberti for pointing out a mistake in the original answer.)

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. These categories were studied in Ulmer's The adjoint functor theorem and the Yoneda embedding (see Theorem 13, for instance), but were not given a name.

The relationship to compactness in the sense of Isbell, and totality, is described in Theorem 8 (where functors preserving all large colimits are called supercocontinuous). In particular, compactness is equivalent to totality, so your notion of strong compactness is a form of "small totality".

(Thanks to Ivan Di Liberti for pointing out a mistake in the original answer.)

Assuming $\mathcal C$ and $\mathcal D$ are locally small and small-cocomplete, a functor $F \colon \mathcal C \to \mathcal D$ is 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. These categories were studied in Ulmer's The adjoint functor theorem and the Yoneda embedding (see Theorem 13, for instance), but were not given a name.

The relationship to compactness in the sense of Isbell, and totality, is described in Theorem 8 (where functors preserving all large colimits are called supercocontinuous). In particular, compactness is equivalent to totality, so your notion of strong compactness is a form of "small totality".

(Thanks to Ivan Di Liberti for pointing out a mistake in the original answer.)