= Solution
For $A\in\mathcal C$, the <comma category> $(A\downarrow G)$ has objects $(D,u)$ with $D\in\mathcal D$ and $u:A\to GD$. A morphism $(D,u)\to(D',u')$ is $h:D\to D'$ satisfying $Gh\,u=u'$. Identities and composition are those of $\mathcal D$, and functoriality of $G$ verifies the condition under composition.
The <Freyd general adjoint functor theorem> states: if $\mathcal D$ is a <locally small category> with all small <categorical limits>, and $\mathcal C$ is locally small, then $G:\mathcal D\to\mathcal C$ has a <left adjoint> if and only if it preserves small limits and satisfies the <solution-set condition>. The latter means that for each $A$ there is a set-indexed family $u_i:A\to GD_i$ such that every $u:A\to GD$ equals $Gh\,u_i$ for some $i$ and some $h:D_i\to D$.
For necessity, use the standard result that a <right adjoint> preserves <categorical limits>. If $F\dashv G$, the singleton family containing the <unit of an adjunction> $\eta_A:A\to GFA$ is a solution set, since transposition gives $u=Gh\,\eta_A$ for a unique $h:FA\to D$.
For sufficiency, use the following standard limit fact: if $\mathcal D$ is complete and $G$ preserves limits, the projection $(A\downarrow G)\to\mathcal D$ creates small limits. Indeed, a compatible family $A\to GD_j$ induces a unique arrow into $G(\lim D_j)$, and this makes the underlying limit a limit in the comma category. The <comma category> is locally small because each of its hom-sets is a subset of a hom-set of $\mathcal D$. Its solution family is a <weakly initial set>.
We prove the remaining <initial-object lemma for complete categories with a weakly initial set>. In any <locally small category> $\mathcal K$ with all small limits and a weakly initial set $(K_i)$, form $W=\prod_iK_i$. It is weakly initial: for any $X$, some $K_i\to X$ exists and may be composed with the projection $W\to K_i$. The empty family cannot be weakly initial in a nonempty complete category, which has a terminal object.
The set $\operatorname{End}(W)$ is small. Form a simultaneous <equalizer> $e:E\to W$ of every endomorphism of $W$ and $1_W$; thus
$$
u e=e\quad\text{for every }u:W\to W.
$$
This equalizer exists by completeness, for example as the equalizer of two maps $W\rightrightarrows W^{\operatorname{End}(W)}$. The object $E$ is still weakly initial, since it maps to $W$.
Given $a,b:E\to X$, take their <equalizer> $j:Y\to E$. Weak initiality of $W$ gives $t:W\to Y$. Since $ejt$ is an endomorphism of $W$, we have $ejte=e$, and cancellation of the <monomorphism> $e$ gives $jte=1_E$. Thus $j$ is a <split epimorphism> as well as a <monomorphism>, so it is an <isomorphism>. From $aj=bj$ follows $a=b$. There is at least one map $E\to X$ by weak initiality, so \b[$E$ is initial].
Apply this lemma to every $(A\downarrow G)$ and choose its <initial object> $(FA,\eta_A)$. For $v:A\to A'$, initiality gives the unique $Fv:FA\to FA'$ satisfying $GFv\,\eta_A=\eta_{A'}v$. Uniqueness proves the functor laws. The same initiality gives natural <bijections>
$$
\mathcal D(FA,D)\cong\mathcal C(A,GD),\qquad h\longmapsto Gh\,\eta_A.
$$
Hence \b[$F\dashv G$], completing the theorem without invoking another adjoint functor theorem.
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