Here is a word that I think should be adopted by the category theorists. (If there is another synonymous word already in existence, please let me know.)

**Definition:** A category $C$ is *saft* if every cocontinuous functor $C \to D$ has a right adjoint.

The word "saft" is an abbreviation for "special adjoint functor theorem (SAFT)", of which there are many, because SAFTs always take the form "If a category $C$ is XYZ, then it is saft." For example: it suffices for a category to be some cocompletion of some small subcategory (F. Ulmer, The adjoint functor theorem and the Yoneda embedding, Illinois Journal of Mathematics, 1971 vol. 15 (3) pp. 355-361).

It is an easy exercise that a category $C$ is saft if and only if every continuous functor $C^{\rm op} \to \text{Set}$ is representable. In particular any saft category is complete, because any putative limit corresponds to a continuous functor, and hence necessarily representable.

On the other hand, I don't see any particular reason why a saft category is necessarily cocomplete, except that every example I know is, and every SAFT uses cocompleteness as one of the conditions.

Question:Is a saft category necessarily cocomplete?