Currying adjunction for small categories ($-\times \mathcal{C}⊣[\mathcal{C},-]$)

A functor $\mathcal{D}\times \mathcal{C}\to \mathcal{E}$ is uniquely a functor $\mathcal{D}\to [\mathcal{C},\mathcal{E}]$: fix its first coordinate and use its first-coordinate arrows as natural transformations. Uncurrying evaluates those transformations and second-coordinate arrows. The adjunction unit inserts a fixed first coordinate, and the adjunction counit is evaluation. This makes the category of small categories cartesian closed.