Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 22 / 3 / b / i

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 $v_{C,x}:C\to A$ satisfiesThe component at $(R,r)$ uniquely determines the mediating morphism. Thus the representing object is the colimit of the elements projection: