= Solution
The <general adjoint functor theorem> has the following limit form. Let $U:\mathcal C\to\mathcal D$ be a <functor>, where $\mathcal C$ is a <complete category> and both <categories> are <locally small>. Then \b[$U$ has a left adjoint exactly when it preserves small limits and satisfies the solution-set condition].
The <solution-set condition> requires that, for each $D\in\mathcal D$, there be a <set> of pairs $(C_i,d_i:D\to UC_i)$ such that every $d:D\to UC$ factors as
$$
\boxed{d=U(h)d_i\quad\text{for some }i\text{ and }h:C_i\to C.}
$$
Equivalently, each <comma category> $(D\downarrow U)$ has a <weakly initial set>. All completeness and preservation requirements here concern small <categorical limits>. Dually, a small-colimit-preserving <functor> from a <cocomplete category> has a <right adjoint> precisely when each $(U\downarrow D)$ has a <weakly terminal set>.
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