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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.

The limit form of the Special adjoint functor theorem is as follows. Let $\mathcal{C}$ be a locally small category, a complete category, and a well-powered category, and suppose it has a small cogenerating family $(Q_i{)}_{i\in I}$. Let $\mathcal{D}$ be a locally small category. Then a functor $G:\mathcal{C}\to \mathcal{D}$ has a left adjoint if and only if it preserves all small categorical limits. The dual exchanges completeness, well-poweredness and cogenerators for cocompleteness, well-copoweredness and generators, and characterizes functors with a right adjoint.

Here a cogenerating set distinguishes unequal parallel maps by postcomposition into one of the $Q_i$. A well-powered category has a set of subobjects of each fixed object. Completeness means existence of all small categorical limits. We use the initial-object lemma for complete categories with a weakly initial set: a complete category that is locally small has an initial object exactly when it has a weakly initial set. A proof is given in Question 5(c). We also use the fact that when $G$ preserves small categorical limits, the comma category $(B\downarrow G)$ is complete, with its limits constructed in $\mathcal{C}$; local smallness is inherited from $\mathcal{C}$. Finally, having an initial object in $(B\downarrow G)$ for each $B$ is the universal arrow from an object to a functor characterization of a left adjoint.

Assume $G$ preserves small categorical limits, and fix $B\in \mathcal{D}$. We will construct a weakly initial set in $(B\downarrow G)$, rather than assume a solution set. Start with an arbitrary object $(C,x)$, where $x:B\to GC$. Call a subobject $m:S\hookrightarrow C$ supporting if $x=G(m)x_S$ for some $x_S:B\to GS$. There is at least one, namely $1_C$. Because $G$ preserves pullbacks in a category, it preserves monomorphisms: the self-pullback characterization of a monic map is preserved. Consequently each $x_S$ is unique.

There is a set of supporting subobjects by well-poweredness. Their intersection of subobjects exists by completeness; write it $m_0:C_0\hookrightarrow C$. Applying $G$ to this intersection categorical limit gives a unique $x_0:B\to G{C}_0$ lifting all the $x_S$, and hence $x=G(m_0)x_0$. Minimality says that any subobject $n:R\hookrightarrow C_0$ supporting $x_0$ is invertible: $m_{0}n$ supports $x$, so $m_0$ factors through $m_{0}n$, giving a right inverse to the monic $n$.

If $u,v:C_0\to Q_i$ satisfy $G(u)x_0=G(v)x_0$, their equalizer supports $x_0$, since $G$ preserves that equalizer. It is therefore an isomorphism, and $u=v$. We have obtained an injective functionLet $J_i$ be its image, a subset of the fixed set $S_i=\mathcal{D}(B,G{Q}_i)$. The evaluation embedding into cogenerator products is a monomorphismIndeed each $s\in J_i$ corresponds to a unique map $C_0\to Q_i$, and the cogenerating set property makes the resulting family jointly monic. Notice that we use the realized subsets $J_i$, not all of $S_i$: there need not be a map $C_0\to Q_i$ available for an unused index.

There is a set of tuples $J=(J_i)$, since $I$ and every $S_i$ are sets. For each resulting $P_J$, choose representatives of its subobjects; these form a set by well-poweredness. For every representative $R$, include every map $y:B\to GR$, a set by local smallness of $\mathcal{D}$. All resulting pairs $(R,y)$ form a set. Our given $(C,x)$ receives a map from such a pair: transport $x_0$ along the isomorphism between $C_0$ and the chosen representative, then compose its inverse with $m_0$. Thus these pairs form a weakly initial set in $(B\downarrow G)$. This is the cogenerator bound for comma-category solution sets.

The initial-object lemma for complete categories with a weakly initial set gives an initial object $(LB,{\eta }_B)$. For $h:B\to B^{\prime }$, define $Lh$ by the unique equationUniqueness proves the functor laws and gives the natural bijectionsThus $L⊣G$. Conversely, a right adjoint preserves small categorical limits: its adjunction identifies maps into the proposed limiting object with compatible families of maps into the diagram. This proves both directions of the limit form of the special adjoint functor theorem. Applying the argument to the opposite categories proves the dual statement. Choices of representatives and adjoint objects are understood in the usual ambient-universe convention for large categories.