 Solution

For $D:\mathcal{I}\to \mathcal{C}$, define the functor $\mathrm{Cone}(-,D):{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$ bywhere $\Delta X$ is the constant diagram functor; a morphism into $X$ acts by precomposition on every categorical cone leg. A categorical limit is a representation of a functor of the formThe image of $1_L$ is the universal categorical cone $\lambda :\Delta L\to D$. By naturality of the representing isomorphism, the image of $h:X\to L$ is $\lambda \circ \Delta h$. Thus this definition is exactly existence and uniqueness of factorization of every categorical cone through $\lambda$. Conversely that universal property gives the representing natural bijection.