Left adjoint in a functor category Edit: Originally put "right adjoint" instead of "left adjoint"; now changed.
If I have small categories $\mathcal{C},\mathcal{D}$ with $F : \mathcal{C} \to \mathcal{D}$ a functor then I want to show that the functor $\tilde{F}:\mathbf{Set}^\mathcal{D} \to \mathbf{Set}^\mathcal{C}$ given by $G \mapsto G \circ F$ has a left adjoint.
It seems like an adjoint functor theorem result but I can't seem to get it to work. I need a weakly initial set in the categories $(H \downarrow \tilde{F})$ for each $H \in \mathbf{Set}^\mathcal{C}$. I was thinking of taking something like $\{(\mathcal{D}(FA,-),??)\}_{A \in \mathcal{C}}$ but I don't know how to get a natural transformation that will work.
Any ideas?
If the $H = \mathcal{C}(A,-)$ then I can get a weakly initial family given by $\{(\mathcal{D}(FA,-),\beta)\}_{A \in \mathcal{C}}$ where $\beta_B(f:A \to B) = Ff \in \mathcal{D}(FA,FB)$, however I don't know how to extend this to other cases. I could add in a choice of $x \in HA$ which would provide a unique natural transformation $\mathcal{C}(A,-) \to H$, however this doesn't seem to help me much.
 A: It seems that you are almost there:
If $\tilde{F}$ has a left adjoint $L$ then we must have
$$
\mathbf{Set}^{\cal D}(L({\cal C}(A,-)),G)
\cong
\mathbf{Set}^{\cal C}({\cal C}(A,-),G\circ F)
\cong
G(F(A))
\cong
\mathbf{Set}^{\cal D}({\cal D}(FA,-)),G)
$$
where the last two isomorphisms come from the Yoneda Lemma.
So this pins down what values $L$ can take on the representable
functors ${\cal C}(A,-)$. In fact, the natural map
$\beta: {\cal C}(A,-) \rightarrow {\cal D}(FA,F-)$
given by $f\mapsto Ff$ already provides the unit at ${\cal C}(A,-)$ of the
possible adjunction (you may need to check this).
Now every functor $H: {\cal C}\rightarrow\mathbf{Set}$ is a colimit
of representable functors. Now, if $L$ is a left adjoint it has to
preserve colimits
$$
L(\mathop{\mathrm{colim}}_i\, {\cal C}(A_i,-))
\cong
\mathop{\mathrm{colim}}_i\, {\cal D}(FA_i,-)
$$
and this extends the previously given 'partial' left adjoint.
For $H=\mathop{\mathrm{colim}}_i\, {\cal C}(A_i,-)$ the unit at $H$ is induced
by the maps
$$
{\cal C}(A_i,-) \stackrel{\beta_i}{\to} {\cal D}(FA_i,F-)
\stackrel{d_i\circ F}{\to} \mathop{\mathrm{colim}}_i {\cal D}(FA_i,-)\circ F
$$
where the $d_i$ are the colimit injections.
