= Solution
Suppose $X\cong\mathcal C(-,R)$. As a <functor> on the <opposite category>, a <representable presheaf> preserves every existing <categorical limit>: a <colimit> in $\mathcal C$ is defined by the <bijection> between <morphisms> from its vertex into $R$ and compatible families of <morphisms> from its diagram objects into $R$. This remains valid for a possibly large <diagram in a category> whenever that <colimit> exists and the compatible-family collection is the corresponding <set>.
The <category of elements> has a <terminal object> $(R,r)$, where $r$ is the image of $1_R$ under the representation. For each $(C,x)$ let $u_{C,x}:C\to R$ be its unique <morphism> to $(R,r)$. These form a <cocone> for the <forgetful functor> $G$. Any competing <cocone> $v_{C,x}:C\to A$ satisfies
$$
v_{C,x}=v_{R,r}\,u_{C,x}.
$$
The component at $(R,r)$ uniquely determines the mediating <morphism>. Thus \b[the representing object is the colimit of the elements projection]:
$$
\boxed{\operatorname{colim}_{(C,x)\in\operatorname{Elts}(X)}C\cong R.}
$$
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