= Solution
The <currying adjunction for small categories> uses the <bijection>
$$
\operatorname{Fun}(\mathcal D\times\mathcal C,\mathcal E)
\cong\operatorname{Fun}(\mathcal D,[\mathcal C,\mathcal E]).
$$
It sends $H$ to the <functor> $d\mapsto H(d,-)$, with an arrow $u:d\to d'$ inducing the <natural transformation> whose $c$-component is $H(u,1_c)$. Conversely, for $K:\mathcal D\to[\mathcal C,\mathcal E]$, define its uncurried <functor> by $(d,c)\mapsto K(d)(c)$ and
$$
(u,v):(d,c)\to(d',c')\quad\longmapsto\quad K(d')(v)K(u)_c.
$$
Naturality of $K(u)$ allows the two factors to be interchanged in the appropriate order, giving functoriality. The constructions are inverse and natural in $\mathcal D$ and $\mathcal E$. Hence $-\times\mathcal C$ is a <left adjoint> to $[\mathcal C,-]$ on the <category of small categories>.
The unit sends $d$ to the <functor> $c\mapsto(d,c)$ and $u$ to the transformation $(u,1_c)$. The counit is evaluation $[\mathcal C,\mathcal E]\times\mathcal C\to\mathcal E$: $(H,c)\mapsto H(c)$ and $(\alpha,v)\mapsto H'(v)\alpha_c=\alpha_{c'}H(v)$.
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