= Solution
For the general alternative, let $G:\mathcal D\to\mathcal C$ be a <functor> between <locally small categories>, with $\mathcal D$ a <complete category>. The <Freyd general adjoint functor theorem> says that $G$ has a <left adjoint> if and only if it preserves small <categorical limits> and satisfies the <solution-set condition>. Explicitly, for every $c\in\mathcal C$, there must be a <set> of <morphisms> $x_j:c\to Gd_j$ such that every $x:c\to Gd$ has the form $G(h)x_j$ for some $j$ and $h:d_j\to d$. Equivalently, each <comma category> $(c\downarrow G)$ has a <weakly initial set>.
We prove the <initial-object lemma for complete categories with a weakly initial set>. Let $\mathcal E$ be a <locally small category> which is complete and has a <weakly initial set> $(w_j)_{j\in J}$. Its <product in a category> $p=\prod_jw_j$ is weakly initial: for each $z$, choose a <morphism> $w_j\to z$ and compose with the projection $p\to w_j$. The <hom-set> $\mathcal E(p,p)$ is a <set>. Form $e:i\to p$ equalizing every <endomorphism> of $p$ with $1_p$, by taking the <equalizer> of the two <morphisms>
$$
p\rightrightarrows\prod_{v\in\mathcal E(p,p)}p,
$$
whose $v$-coordinates are $v$ and $1_p$. Then $ve=e$ for every <endomorphism> $v$, and $e$ is monic. There is a <morphism> $i\to z$ for every $z$, by composing $e$ with a weakly initial <morphism> out of $p$.
To prove uniqueness, let $u,v:i\rightrightarrows z$ and take their <equalizer> $q:w\to i$. Weak initiality of $p$ gives $a:p\to w$. The composite $eqa:p\to p$ is an <endomorphism>, so $eqa e=e$. Cancel the <monomorphism> $e$ to obtain $qa e=1_i$. Thus $q$ is both monic and split epic, hence an <isomorphism>: if $qk=1$, cancellation from $qkq=q$ gives $kq=1$. Since $uq=vq$, it follows that $u=v$. Therefore $i$ is an <initial object>, proving the lemma fully.
Now assume <categorical limit> preservation and the <solution-set condition>. The <comma category> $(c\downarrow G)$ is locally small. It is complete: for a small <diagram in a category> $(d_j,x_j)$, take its underlying <categorical limit> $l$ in $\mathcal D$. Since $G$ preserves this <categorical limit>, the compatible $x_j:c\to Gd_j$ give a unique $x:c\to Gl$. The same <universal property> shows that $(l,x)$ is the <categorical limit> in the <comma category>. Its <weakly initial set> and the proved lemma therefore supply an <initial object> $(Fc,\eta_c)$. For $t:c\to c'$, its <universal property> defines a unique $Ft:Fc\to Fc'$ with $G(Ft)\eta_c=\eta_{c'}t$. Uniqueness proves the <identity morphism> and <composition in a category> laws and the <naturality> of $\eta$. The same <universal property> gives the <natural bijection> $\mathcal D(Fc,d)\cong\mathcal C(c,Gd)$, so $F\dashv G$.
For necessity, an <adjunction> $F\dashv G$ supplies the singleton <weakly initial set> $(Fc,\eta_c)$ at $c$. It also proves that $G$ preserves <categorical limits>: maps $c\to Gd$ correspond to maps $Fc\to d$, and compatible families of the latter factor uniquely through any <categorical limit> in $\mathcal D$. Hence the image <categorical cone> under $G$ has the required <universal property>. This proves both directions of the general alternative.
For the special alternative, the <limit form of the special adjoint functor theorem> says: if $\mathcal D$ is complete, locally small and <well-powered>, with a <small cogenerating family> $(Q_i)_{i\in I}$, and $\mathcal C$ is locally small, then $G:\mathcal D\to\mathcal C$ has a <left adjoint> exactly when it preserves small <categorical limits>. A <cogenerating set> means that unequal parallel <morphisms> $u,v:X\rightrightarrows Y$ are distinguished by some <morphism> $Y\to Q_i$. We prove the needed <solution-set condition>, so that the general theorem applies.
Fix $x:c\to Gd$. A supporting <subobject> of $d$ is a <monomorphism> $m:d'\to d$ through whose image under $G$ the <morphism> $x$ factors. There is at least one, namely $1_d$. Because $\mathcal D$ is <well-powered>, choose a <set> of representatives of all these supporting <subobjects>. A limit-preserving <functor> preserves <monomorphisms>: a map is monic exactly when its diagonal into its self-<pullback in a category> is an <isomorphism>, and both the <pullback in a category> and this <isomorphism> are preserved. Consequently each factorization of $x$ through $Gm$ is unique. Form the <intersection of subobjects> of the supporting representatives, as a wide <pullback in a category> over $d$. The unique lifts of $x$ are compatible and, since $G$ preserves that <categorical limit>, they give a factorization
$$
x=G(m_0)x_0,\qquad m_0:d_0\hookrightarrow d,\quad x_0:c\to Gd_0.
$$
This is a <minimal supported subobject>. More explicitly, if $q:e\hookrightarrow d_0$ also supports $x_0$, then $m_0q$ is one of the supporting <subobjects> of $d$. The intersection therefore factors through $m_0q$, giving $r:d_0\to e$ with $m_0qr=m_0$. Monicity of $m_0$ gives $qr=1_{d_0}$; monicity of $q$ then makes $q$ invertible.
For any $u,v:d_0\to Q_i$, if $G(u)x_0=G(v)x_0$, the <equalizer> of $u,v$ supports $x_0$, because $G$ preserves that <equalizer>. Minimality makes it invertible, so $u=v$. We have therefore obtained <injective functions>
$$
\mathcal D(d_0,Q_i)\longrightarrow\mathcal C(c,GQ_i),\qquad u\longmapsto G(u)x_0.
$$
Let $S_i$ be the realized image <subset> of the fixed <set> $\mathcal C(c,GQ_i)$. Use the unique <morphism> corresponding to each $s\in S_i$ to form the <evaluation embedding into cogenerator products>
$$
d_0\longrightarrow P_S=\prod_{i\in I}\prod_{s\in S_i}Q_i.
$$
This <morphism> is monic: if two <morphisms> into $d_0$ have identical product composites, every <morphism> from $d_0$ to every $Q_i$ gives identical composites, and the cogenerating property makes the two <morphisms> equal. There is only a <set> of possible families of <subsets> $(S_i)$, and hence a <set> of possible products $P_S$. Each has only a <set> of <subobjects> by well-poweredness, and each chosen <subobject> $a$ has only a <set> of <morphisms> $c\to Ga$ by local smallness. Take all these pairs $(a,z:c\to Ga)$ as a <weakly initial set>. Every original $(d,x)$ receives a <morphism> from a pair representing its $(d_0,x_0)$, through $m_0$. Thus the <set> is weakly initial in $(c\downarrow G)$.
The general theorem now supplies a <left adjoint>. Necessity again follows from preservation of <categorical limits> by a <right adjoint>. This proves the special alternative, including its size argument. Using only the realized <subsets> $S_i$ avoids assuming that an arbitrary object admits a <morphism> into every cogenerator. \b[Both adjoint <functor> theorem alternatives are proved.]
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