Iterated small limits commute: either order is canonically isomorphic to the limit over the product category

By [F2], a cone from $X$ to $D$ is exactly a family of arrows $X\to D(j,k)$ compatible separately with every $\mathcal{J}$-arrow and every $\mathcal{K}$-arrow.

For each $j$, the $\mathcal{K}$-limit turns such a compatible $k$-family into one unique arrow $X\to {\lim }_{k}D(j,k)$. Compatibility in $j$ turns these into a cone over the resulting $\mathcal{J}$-diagram, and its limit turns the family into one unique arrow $X\to {\lim }_j{\lim }_{k}D(j,k)$. Both constructions reverse by the two universal properties.

Hence the first iterated limit has the universal property of the $\mathcal{J}\times \mathcal{K}$-limit. By [L1] it is uniquely compatibly isomorphic to that limit. Interchanging $j$ and $k$ proves the second isomorphism.

If either index category is empty, step 1.1 describes an empty family, so each existing expression is a terminal object and [L1] gives the same canonical isomorphisms. The morphisms of $\mathcal{J}\times \mathcal{K}$ form a subset of the Cartesian product of the two morphism sets, so [F3] makes the product category small and no large diagram has been introduced.