Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 18 / 3 / d

Let $U\cong \mathcal{C}(R,-)$ be a representable functor, and let $(L,p_j)$ be the categorical limit of a small diagram $D:J\to \mathcal{C}$. A morphism $R\to L$ is uniquely equivalent to a family $f_j:R\to D(j)$ satisfying $D(u)f_j=f_k$ for every $u:j\to k$.

Such compatible families are exactly the elements of the categorical limit of the set-valued diagram $\mathcal{C}(R,D(-))$. Consequently the canonical comparisonis a bijection. Transporting it through the representing natural isomorphism proves that $U$ preserves every small limit that exists in $\mathcal{C}$. This proves that covariant representables preserve limits. For an empty diagram this says that maps into a terminal object form a singleton.