= Solution
For a small diagram $F:I\to\mathcal D$ in a <locally small category>, define its cone <functor> on $\mathcal D^{\mathrm{op}}$ by
$$
\operatorname{Cone}_F(A)=\operatorname{Nat}(\Delta A,F)\cong\lim_{i\in I}\mathcal D(A,F(i)),
$$
where $\Delta A$ is the constant diagram. Precomposition acts on cone vertices. A <categorical limit> is a representation of this <functor>: an object $L$ with natural bijections $\mathcal D(A,L)\cong\operatorname{Cone}_F(A)$. 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.
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