 Solution

For a small diagram $F:I\to \mathcal{D}$ in a locally small category, define its cone functor on ${\mathcal{D}}^{\mathrm{op}}$ bywhere $\Delta A$ is the constant diagram. Precomposition acts on cone vertices. A categorical limit is a representation of this functor: an object $L$ with natural bijections $\mathcal{D}(A,L)\cong {\mathrm{Cone}}_F(A)$. The image of $1_L$ is the universal cone, and the inverse bijection supplies the unique mediating arrow for every cone. Thus the representable functor formulation and the usual terminal-cone universal property coincide.