= Solution
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 comparison
$$
\mathcal C(R,L)\longrightarrow\lim_{j\in J}\mathcal C(R,D(j)),\qquad f\longmapsto(p_jf)_j,
$$
is a <bijection>. Transporting it through the representing <natural isomorphism> proves that \b[$U$ 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.
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