 Solution

The currying adjunction for small categories uses the bijectionIt sends $H$ to the functor $d\mapsto H(d,-)$, with an arrow $u:d\to d^{\prime }$ 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 $(d,c)\mapsto K(d)(c)$ andNaturality 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 $(\alpha ,v)\mapsto H^{\prime }(v){\alpha }_c={\alpha }_{c^{\prime }}H(v)$.