 Limit form of the special adjoint functor theorem

A functor $G:\mathcal{C}\to \mathcal{D}$ from a locally small category that is complete and well-powered, with a small cogenerating family, to a locally small category has a left adjoint exactly when it preserves small categorical limits. For sufficiency, the cogenerator bound for comma-category solution sets produces a weakly initial set in each complete comma category $(B\downarrow G)$. The initial-object lemma for complete categories with a weakly initial set supplies its initial object, a universal arrow from an object to a functor. Necessity is limit preservation by a right adjoint.