Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 17 / 7 / i / Solution

If $F⊣U$, let $1$ denote a singleton set. The adjunction gives the natural bijectionsThus $U$ is represented by $F1$, proving $A\Rightarrow R$.

If $U\cong \mathcal{C}(R,-)$ is a covariant representable functor, a morphism $R\to {\lim }_j{C}_j$ is exactly a compatible family of morphisms $R\to C_j$. Consequently the comparison mapis a bijection for every existing small categorical limit. This proves $R\Rightarrow L$ by the universal property of a limit. In particular,