Ends exist over a small index category in a complete target, and coends in a cocomplete one

Let $\mathcal{C}$ be a small category (Small, locally small, and large categories) and let $T:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$ be a functor.

If $\mathcal{D}$ is complete, then $T$ has an end. If $\mathcal{D}$ is cocomplete, then $T$ has a coend (Finite, small, and large limits and colimits; complete and cocomplete categories, The end and the coend of a functor ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$).

These conditions are sufficient and are not asserted to be necessary: the definition of an end asks only that a terminal wedge exist, and a particular functor on a large $\mathcal{C}$, or into a target that is not complete, may still have one.

The smallness count is carried out rather than asserted because it is where a size hypothesis could quietly be dropped: it is the morphisms of $\mathrm{Tw}(\mathcal{C})$, not only its objects, that have to form a set before $T\pi$ counts as a small diagram, and that in turn needs $\mathrm{Mor}(\mathcal{C})$ to be a set rather than merely each hom-set to be one. A locally small but large $\mathcal{C}$ is not enough.

Sufficiency is all that is claimed. That the hypotheses cannot simply be dropped is FALSE: every functor on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ has an end, which exhibits a small index category and a target that is not complete in which an end fails to exist.