Solution-set condition

For $U:\mathcal{C}\to \mathcal{D}$, the left-adjoint solution-set condition requires a weakly initial set in every $(D\downarrow U)$. Equivalently, there is a set of arrows $d_i:D\to U{C}_i$ through which every $d:D\to UC$ factors as $U(h)d_i$. The dual right-adjoint condition asks for weakly terminal solution sets in $(U\downarrow D)$.