Does $\mathrm{Fun}$ preserve adjunctions?

Question

Let $u: A \to B$ be a left adjoint functor with right adjoint $v: B \to A$ and let $C$ be a further category. Is it true that $\mathrm{Fun}(C, u)$ is left adjoint to $\mathrm{Fun}(C, v)$? Similar question for $\mathrm{Fun}(u, C)$ and $\mathrm{Fun}(v, C)$.

Answer 1

Zhen Lin has given the answer in the comments, but it could be helpful to spell out further. $\mathbf{Fun} \colon \mathbf{Cat}° \times \mathbf{Cat} \to \mathbf{Cat}$ is a 2-functor. It is a general fact that 2-functors preserve adjunctions (since adjunctions are diagrammatic notions, via the $\eta$–$\epsilon$ formulation). Therefore, fixing the first variable, we have that $\mathbf{Fun}(C, {-}) \colon \mathbf{Cat} \to \mathbf{Cat}$ preserves adjunctions; and fixing the second variable, we have that $\mathbf{Fun}({-}, C) \colon \mathbf{Cat}° \to \mathbf{Cat}$ preserves adjunctions. However, the latter sends adjunctions in $\mathbf{Cat}°$ to adjunctions in $\mathbf{Cat}$, which has the effect of swapping the left and right adjoint.