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A precise commutation of iterated categorical limits statement is the following. For small categories $\mathcal{I},\mathcal{J}$ and $F:\mathcal{I}\times \mathcal{J}\to \mathcal{D}$, if every column categorical limit and the resulting $\mathcal{J}$-limit exist, thenIf every row categorical limit and the resulting $\mathcal{I}$-limit also exist, thenEach isomorphism is the unique one identifying all projections to $F(i,j)$. Indeed, the preceding part proves that each iterated categorical limit has the universal property of the whole product diagram in a category. Its uniqueness gives the canonical isomorphism. Equivalently, categorical cones with vertex $X$ over the product diagram in a category are exactly compatible families, indexed by $j$, of categorical cones over $F(-,j)$; regrouping the same compatibility equations cannot change their representing object. In a complete category all the stated existence hypotheses hold for small $\mathcal{I},\mathcal{J}$. The theorem is about categorical limits with categorical limits, and makes no unconditional claim about commuting a categorical limit with a colimit.

The categorical commutation of iterated categorical limits theorem is the following. For small categories $I,J$ and a two-variable functor $F:I\times J\to \mathcal{D}$, if the two collections of inner limits and their outer limits exist, then there are canonical isomorphismsIn particular all these limits exist if $\mathcal{D}$ has all $I$-shaped and all $J$-shaped limits. The isomorphisms are natural in $F$ and uniquely characterized by preserving every projection to $F(i,j)$.

Part (iii) proves that each iterated limit represents the functor of cones on the product diagram, with the same flattened projections. Two such representations have unique inverse cone-compatible arrows, proving the isomorphisms. For a natural transformation $F\to F^{\prime }$, the two composites in any naturality square have the same product-diagram projections, hence agree by limit uniqueness. This proves naturality as well. Completeness of $\mathcal{D}$ is a convenient sufficient hypothesis, rather than a necessary one for this individual diagram.

Regard the bifunctor as $D^{(2)}:K\to [J,\mathcal{C}]$. Its pointwise categorical limit $E={\lim }_k{D}^{(2)}k$ satisfieswith its action on morphisms obtained by the construction in part (a). Applying the limit-preservation result from part (d) givesThe comparison isomorphism is canonical relative to the chosen categorical limit cones: it is the unique morphism identifying all the projections to $D(j,k)$. This proves commutation of iterated categorical limits, including an empty $J$ or $K$. Equivalently, both iterated constructions have the universal property of the categorical limit of $D$ on $J\times K$.