= Solution
Suppose $G$ is a <representable functor>, with a <natural isomorphism> $\theta:\mathcal C(R,-)\to G$. Put $r=\theta_R(1_R)$. For any $(A,a)$ in the <comma category> $(1\downarrow G)$, the representing <bijection> gives a unique $f:R\to A$ with $\theta_A(f)=a$. By <naturality>, $\theta_A(f)=G(f)(r)$, exactly the condition for $f:(R,r)\to(A,a)$. Hence $(R,r)$ is an <initial object>.
Conversely, if $(R,r)$ is an <initial object>, then
$$
\boxed{\theta_A:\mathcal C(R,A)\longrightarrow G(A),\qquad f\longmapsto G(f)(r)}
$$
is a <bijection> for each $A$: existence and uniqueness are precisely initiality applied to every $(A,a)$. For $h:A\to B$, $G(h)\theta_A(f)=G(hf)(r)=\theta_B(hf)$, proving <naturality>. Thus $r$ is a <universal element> and $\theta$ is a <representation of a functor>. \b[Being <representable> is equivalent to the singleton <comma category> having an <initial object>.]
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