Orientation and notation conventions in force on this page

Three conventions are in force throughout this page, each of which is reversed by some part of the literature.

Integral signs. The subscripted integral denotes the end and the superscripted integral the coend (The end and the coend of a functor ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$): ${\int }_{c}T(c,c)$ is the terminal wedge and ${\int }^{c}T(c,c)$ the initial cowedge. The variable is bound in both.

The twisted arrow category. $\mathrm{Tw}(\mathcal{C})$ has the morphisms of $\mathcal{C}$ as objects and a morphism $f\to g$ is a pair $(a,b)$ with $bfa=g$, so its projection lands in ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ (The twisted arrow category and its projection to ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$, Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$).

Which category computes a coend. Under those two conventions an end is a limit over $\mathrm{Tw}(\mathcal{C})$ and a coend is a colimit over $\mathrm{Tw}(\mathcal{C}{)}^{\mathrm{o}\mathrm{p}}$ of the integrand read with the domain and codomain of each arrow interchanged (An end is a limit over the twisted arrow category, and a coend is a colimit over its opposite).

Every one of the three is a choice, and the alternative choice is in print.

The reversal of the integral signs is not hypothetical: Yoneda's 1960 paper calls integration what is now the coend and writes it with a subscript, and the opposite convention to the one above is used by some modern authors as well. Anyone converting a formula from another source must check which convention that source fixed before comparing it with a formula on this page; the mathematics is unaffected and only the symbols move.

The orientation of the twisted arrow category is reversed by some sources, so that what is written $\mathrm{Tw}(\mathcal{C})$ there is $\mathrm{Tw}(\mathcal{C}{)}^{\mathrm{o}\mathrm{p}}$ here. Under the reversed orientation the projection lands in $\mathcal{C}\times {\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ and every statement on this page naming $\mathrm{Tw}(\mathcal{C})$ has to be read with the opposite category substituted.

The third convention is a consequence of the first two rather than an independent choice, and it is the one that is easiest to get wrong, because the index category for the coend is the opposite of the index category for the end and the integrand is reindexed. Taking only one of the two changes gives a different object, and a false statement recording exactly that failure is carried on this page.