= Solution
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> \b[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 $x_0:B\to GC_0$ 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: $m_0n$ supports $x$, so $m_0$ factors through $m_0n$, 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 function>
$$
\mathcal C(C_0,Q_i)\longrightarrow\mathcal D(B,GQ_i),\qquad u\longmapsto G(u)x_0.
$$
Let $J_i$ be its image, a subset of the fixed <set> $S_i=\mathcal D(B,GQ_i)$. The <evaluation embedding into cogenerator products> is a <monomorphism>
$$
C_0\hookrightarrow P_J:=\prod_{i\in I}\prod_{s\in J_i}Q_i.
$$
Indeed 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 $h:B\to B'$, define $Lh$ by the unique equation
$$
G(Lh)\eta_B=\eta_{B'}h.
$$
Uniqueness proves the <functor> laws and gives the <natural bijections>
$$
\boxed{\mathcal C(LB,C)\cong\mathcal D(B,GC),\qquad f\longmapsto G(f)\eta_B.}
$$
Thus $L\dashv G$. 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>.
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