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

Choose a limit $L_j$ for each diagram $F(-,j)$, with legs $p_{i,j}:L_j\to F(i,j)$. For an arrow $v:j\to j^{\prime }$, the maps $F(1_i,v)p_{i,j}$ form a cone over $F(-,j^{\prime })$, since the two-variable functor respects the commuting arrows in $I\times J$. The limit property therefore gives a unique arrow $L(v):L_j\to L_{j^{\prime }}$ satisfyingThe arrows $L(1_j)$ and $1_{L_j}$ have identical composites with all projections, so they are equal. The same uniqueness shows $L(v^{\prime }v)=L(v^{\prime })L(v)$. Hence the assignment extends to a functor $L:J\to \mathcal{D}$.

For the chosen limit objects and cones it is uniquely determined, literally, by these equations. Changing the choices gives a unique family of isomorphisms between the limits compatible with their projections; the equations above show that this family is a natural isomorphism of the resulting functors. Thus it is unique up to a unique cone-compatible natural isomorphism, not independent of choices as an equality of objects.