Solution Set Condition

A functor $F:\mathcal{C}\to\mathcal{D}$ satisfies the solution set condition if for every $X:\mathcal{D}$, the Comma Categories $X\downarrow F$ have a Weakly Initial Family of objects.

Intuition

The point of the solution set condition is to be able to replace a potentially overambitous (co)limit that would only exist in a Poset with a smaller one that results in the same (co)limit.

This means that there are many possible solution set conditions; see the Knaster-Tarski Theorem for Categories for another such example.