= Solution
Use the <Freyd general adjoint functor theorem>: a <functor> $U:\mathcal A\to\mathcal B$, with $\mathcal A$ <complete> and both categories <locally small>, has a <left adjoint> precisely when it preserves small <categorical limits> and satisfies the <solution-set condition>. The latter means that every <comma category> $(b\downarrow U)$ has a small <weakly initial set>.
Here is the proof. First prove the <initial-object lemma for complete categories with a weakly initial set>. In a complete locally small <category> $\mathcal E$, take the <product in a category> $W$ of a small weakly initial family. It is weakly initial: project to whichever member admits a <morphism> into the required object. Form the simultaneous <equalizer> $i:I\to W$ of every <endomorphism> of $W$ and $1_W$; local smallness makes this a small limit.
For parallel $f,g:I\to X$, let $j:Y\to I$ be their <equalizer>. Weak initiality supplies $t:W\to Y$. Since $ijt$ is an endomorphism of $W$, the defining equalities give $ijti=i$. Cancel the <monomorphism> $i$ to get $jti=1_I$. Thus $j$ is a split epimorphism as well as a monomorphism, hence invertible, and $f=g$. Existence of a <morphism> $I\to X$ follows by composing $i$ with a map $W\to X$. Therefore $I$ is an <initial object>.
Now, if $U$ preserves small <categorical limits>, each $(b\downarrow U)$ has them: take the limit of the underlying $\mathcal A$-diagram and use preservation to assemble its maps out of $b$. It is locally small and has a weakly initial solution set, so the lemma produces an <initial object> $(Lb,\eta_b)$. Its <universal property> gives natural <bijections>
$$
\mathcal A(Lb,a)\cong\mathcal B(b,Ua).
$$
The universal arrows define $L$ on <morphisms>, proving $L\dashv U$. Conversely, a <right adjoint> preserves <categorical limits>, and its unit supplies a singleton solution set in each comma category.
Apply this to $U=F^*$. The <functor category> $[\mathcal D,\mathbf{Set}]$ is complete and locally small because $\mathcal D$ is small; <pointwise limits in a functor category> show $F^*$ preserves all small limits. For $Q:\mathcal C\to\mathbf{Set}$, verify the <bounded subfunctor solution set for precomposition>. Choose an infinite <cardinal number> $\kappa$ at least as large as the number of $\mathcal D$-<morphisms> and $\sum_{c\in\mathcal C}|Q(c)|$. Given $\eta:Q\to GF$, set
$$
G'(d)=\{G(u)\eta_c(x):c\in\mathcal C,\ x\in Q(c),\ u:Fc\to d\}.
$$
Postcomposition makes $G'$ a subfunctor, each value has cardinality at most $\kappa$, and $\eta$ factors through $G'F$. After labelling its values by subsets of $\kappa$, only a set of possibilities for $G'$ and $Q\to G'F$ exists. Their maps into $G$ give the required weakly initial set. Thus \b[$F^*$ has a left adjoint].
It is the <left Kan extension>. The <left Kan extension as a comma-category colimit> makes it concrete:
$$
\boxed{(\operatorname{Lan}_FQ)(d)=\operatorname{colim}_{(c,u:Fc\to d)\in(F\downarrow d)}Q(c).}
$$
Every representative $(c,u,x)$ maps under $\eta:Q\to GF$ to $G(u)\eta_c(x)$. The colimit identifications and naturality make this well-defined, giving exactly $\operatorname{Nat}(\operatorname{Lan}_FQ,G)\cong\operatorname{Nat}(Q,GF)$.
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