I was studying the SAFT theorem (Special Adjoint Functor Theorem), using Leinster's Basic Category Theory and I had the homework to dualize it. So far I have the following,
Consider a category $\mathscr{A}$, a co-complete category $\mathscr{B}$ and a functor $G: \mathscr{B} \rightarrow \mathscr{A} $. Suppose that $\mathscr{A},\mathscr{B}$ are locally small and $\mathscr{B}$ has a generator and $\mathscr{B}$ is ???. Then $$ \text{G has a right adjunction} \Leftrightarrow \text{G preserves colimits} \\ $$
Where I wrote ?? I am looking for the dual concept of a Well-Powered Category. But I have no idea how to construct it, can someone help me? Or give me an idea of how to? I know that we say that a category is a well powered if every object has a small poset of subobjects.
Any help is appreciated!