Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 18 / 3 / iii

Let $M={\lim }_{j\in J}{L}_j$ with projections $q_j:M\to L_j$. The composites $p_{i,j}{q}_j$ form a cone over $F:I\times J\to \mathcal{D}$: the $I$ equations hold in each inner limit, and the $J$ equations follow from the naturality of $p$ and the outer cone.

Given any cone $a_{i,j}:A\to F(i,j)$, the inner limit property gives unique $b_j:A\to L_j$ with $p_{i,j}{b}_j=a_{i,j}$. The $J$ cone equations for $a$ force $L(v)b_j=b_{j^{\prime }}$ by uniqueness of inner mediating arrows. Thus the $b_j$ form a cone to $L$, and there is a unique $b:A\to M$ with $q_{j}b=b_j$. This factors all $a_{i,j}$, and the two stages of uniqueness prove uniqueness for the whole product diagram. Consequently