 Solution

We prove both versions. In the general adjoint functor theorem, let $G:\mathcal{D}\to \mathcal{C}$ be a functor between locally small categories, with $\mathcal{D}$ complete. Then $G$ has a left adjoint if and only if it preserves small limits and satisfies the solution-set condition: for each $C\in \mathcal{C}$ there is a set of arrows $u_i:C\to G{D}_i$ such that every arrow $u:C\to GD$ factors as $G(h)u_i$ for some $i$ and $h:D_i\to D$.

First prove the initial-object lemma for complete categories with a weakly initial set. Let $\mathcal{A}$ be locally small and complete, and have a small weakly initial family $(W_i)$. Its product $W$ is weakly initial: a map $W_i\to A$ can be composed with the product projection. Take $e:E\to W$ to be the simultaneous equalizer of every endomorphism of $W$ and its identity. This is a small limit since $\mathcal{A}(W,W)$ is a set. Maps $E\to A$ exist by composing $e$ with a weakly initial map $W\to A$.

For uniqueness, take any $a,b:E\to A$ and their equalizer $j:Z\to E$. Weak initiality of $W$ gives $t:W\to Z$. The endomorphism $ejt$ of $W$ fixes $e$ by its defining equalizer property, so $ejte=e$. Since $e$ is monic, $jte=1_E$. Thus $j$ is both monic and split epic, hence invertible, forcing $a=b$. Therefore $E$ is initial. This proves the lemma without an unproved initial-object assertion.

Now fix $C\in \mathcal{C}$. The comma category $(C\downarrow G)$ is locally small. It is complete: take the limit of the underlying diagram in $\mathcal{D}$, use preservation by $G$ to factor the compatible arrows from $C$, and obtain the universal comma cone. The solution set is exactly a small weakly initial family in this comma category. The lemma provides its initial object $(FC,{\eta }_C)$. Its universal property is the natural bijection $\mathcal{D}(FC,D)\cong \mathcal{C}(C,GD)$. For $v:C\to C^{\prime }$, initiality gives the unique $Fv$ satisfying $G(Fv){\eta }_C={\eta }_{C^{\prime }}v$; uniqueness proves identities and composition. Thus these objects form a left adjoint. Conversely a left adjoint provides the singleton solution set consisting of its unit arrow, and a right adjoint preserves limits: the adjunction bijections identify cones into the image diagram with cones from the corresponding left-adjoint object into the original diagram. This proves both directions of the general theorem.

The limit form of the special adjoint functor theorem assumes instead that $\mathcal{D}$ is complete, locally small and well-powered, and has a small cogenerating family $(Q_i)$. For any locally small target, a functor $G:\mathcal{D}\to \mathcal{C}$ has a left adjoint exactly when it preserves small limits. Necessity is already proved. For sufficiency we construct a solution set, so that the general theorem applies.

Fix $C$. Given $u:C\to GD$, consider all pairs $h,k:D\to Q_i$ with $G(h)u=G(k)u$. There is only a set of these pairs. Intersect their equalizers to obtain a monomorphism $m:D^{\prime }\to D$. Since $G$ preserves this limit and $u$ equalizes the images of all pairs, it factors uniquely as $u=G(m)u^{\prime }$ for $u^{\prime }:C\to G{D}^{\prime }$.

For each realized pair $(i,v)$ with $v=G(h)u:C\to G{Q}_i$, choose one such $h$. Any other $k$ with the same pair restricts to the same map on $D^{\prime }$. The restricted chosen maps jointly separate morphisms into $D^{\prime }$: if two such morphisms become equal after all chosen maps, they become equal after every $hm$, and the cogenerating family then makes their composites with $m$ equal; monicity of $m$ cancels it. Hence the resulting mapis monic, where $J$ is a subset of the fixed set ${\coprod }_i\mathcal{C}(C,G{Q}_i)$.

There is only a set of possible $J$, a set of subobjects of each corresponding product by well-poweredness, and a set of arrows from $C$ into $G$ of each chosen subobject representative by local smallness of $\mathcal{C}$. Collect all these representative comma objects. The factorization just constructed shows that they form a weakly initial set in $(C\downarrow G)$. This is the cogenerator bound for comma-category solution sets, with realized subsets avoiding any assumption that all proposed maps to cogenerators have lifts. The general theorem now gives the left adjoint and proves the special theorem.