= Solution
We use the <general adjoint functor theorem> in its solution-set form. Let $G:\mathcal D\to\mathcal C$ be a <functor> between <locally small categories>, with $\mathcal D$ a <complete category>. Then $G$ has a <left adjoint> exactly when it preserves all small <categorical limits> and, for each $C\in\mathcal C$, there is a set-indexed family
$$
u_i:C\to GD_i\qquad(i\in I_C)
$$
such that every $u:C\to GD$ factors as $Gf\,u_i$ for some $i$ and some $f:D_i\to D$. This is the <solution-set condition>. The indexing family is a set, whereas the entire <comma category> $(C\downarrow G)$ can be large.
We first prove the <initial-object lemma for complete categories with a weakly initial set>. Let $\mathcal E$ be locally small and complete, with a <weakly initial set> $(E_i)_{i\in I}$. Its <product in a category> $W=\prod_iE_i$ is weakly initial: follow a projection by a map from the corresponding $E_i$ to any desired object. The <hom-set> $\mathcal E(W,W)$ is a set, so we can form the simultaneous <equalizer>
$$
e:E\longrightarrow W,\qquad he=e\quad\text{for every }h:W\to W.
$$
This is a small limit; equivalently, equalize the family of endomorphisms with the family of identity maps into a product of copies of $W$. Every object receives a map from $E$, since it receives one from $W$.
To prove uniqueness, let $a,b:E\rightrightarrows B$, and take their equalizer $j:V\to E$. Weak initiality of $W$ gives $t:W\to V$. Since $ejt$ is an endomorphism of $W$, the defining property of $e$ gives
$$
ejte=e.
$$
The <monomorphism> $e$ cancels, yielding $jte=1_E$. Thus $j$ is both a monomorphism and a <split epimorphism>, hence an <isomorphism>. Its equalizer property now implies $a=b$. Therefore \b[$E$ is initial].
Fix $C$. Small <categorical limits> in $(C\downarrow G)$ are obtained from the limits in $\mathcal D$: preservation by $G$ supplies the unique map from $C$ to the image of the limiting object. The <comma category> is locally small, and the solution family is a <weakly initial set> in it. The lemma supplies an <initial object> $(LC,\eta_C:C\to GLC)$.
For $k:C\to C'$, initiality gives a unique arrow $Lk:LC\to LC'$ satisfying $G(Lk)\eta_C=\eta_{C'}k$. Uniqueness proves preservation of identities and composition, so $L$ is a <functor>. Initiality also gives <natural bijections>
$$
\boxed{\mathcal D(LC,D)\cong\mathcal C(C,GD),\qquad f\longmapsto Gf\,\eta_C,}
$$
which prove $L\dashv G$.
Conversely, if $L\dashv G$, the singleton $\eta_C:C\to GLC$ is a solution set by the <adjunction> bijection. To see that $G$ preserves a small limit $D=\lim_jD_j$, use the adjunction and the limit property to obtain, naturally in $C$,
$$
\mathcal C(C,GD)\cong\mathcal D(LC,D)\cong\lim_j\mathcal D(LC,D_j)\cong\lim_j\mathcal C(C,GD_j).
$$
This is precisely the <universal property> of the image limit cone. It proves the necessity of both hypotheses.
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