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.

Under the hypotheses of the limit form of the special adjoint functor theorem, intersect all subobjects of $C$ supporting $x:B\to GC$. Limit preservation gives a smallest supporting $(C_0,x_0)$. If $G(u)x_0=G(v)x_0$ for maps $u,v:C_0\to Q_i$, their equalizer supports $x_0$, so minimality makes it invertible and $u=v$. Thus the indicated map of hom-sets is injective. The evaluation embedding into cogenerator products embeds $C_0$ in a product of cogenerators indexed by the realized subsets of the fixed sets $\mathcal{D}(B,G{Q}_i)$. There are only a set of such products, a set of their subobjects, and a set of maps from $B$ into their images under $G$. These data give a weakly initial set in $(B\downarrow G)$. Using realized subsets avoids assuming maps into every cogenerator exist.