Commutation of iterated categorical limits (${\lim }_j{\lim }_{k}D(j,k)\cong {\lim }_k{\lim }_{j}D(j,k)$)

For small $J,K$ and a functor $D:J\times K\to \mathcal{C}$ into a complete category, the two iterated categorical limits are canonically isomorphic. They have the same projections to every $D(j,k)$ and the same universal property. The result follows also from preservation of categorical limits by the limit functor.