Iterated ends may be taken in either order

Let $T:(\mathcal{C}\times \mathcal{D}{)}^{\mathrm{o}\mathrm{p}}\times (\mathcal{C}\times \mathcal{D})\to \mathcal{E}$ be a functor, reindexed as in Fubini: an end over a product index category and the two iterated ends exist together and agree, and assume chosen families of inner ends in each order together with the functor structures in their remaining parameters (Ends and coends with parameters, Product category and its projection functors).

If either iterated end exists, so does the other, and there is exactly one isomorphism

${\int }_c{\int }_{d}T(c,d,c,d)\cong {\int }_d{\int }_{c}T(c,d,c,d)$

commuting with every component of the two wedges over the product index category that they induce (The end and the coend of a functor ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$). The hypotheses are exactly those of Fubini: an end over a product index category and the two iterated ends exist together and agree and nothing is added.

The corollary is a statement about two objects that are each characterised by a universal property, so the isomorphism it produces is canonical in the strong sense: it is determined by the requirement that it commute with the components, and no choice is involved beyond the two chosen families of inner ends already assumed by Fubini: an end over a product index category and the two iterated ends exist together and agree.

Nothing here says that either iterated end exists. What makes the interchange usable in practice is a separate existence statement, such as Ends exist over a small index category in a complete target, and coends in a cocomplete one applied to each inner integrand.