For the direct limit of a sequence of ultrapowers, see Ultraproduct.

Take now the direct limit of the 's.

At limit stages, such as "A", form the direct limit of earlier stages.

The direct limit can be defined in an arbitrary category by means of a universal property.

The direct limit of this system is an object in together with morphisms satisfying .

Unlike for algebraic objects, the direct limit may not exist in an arbitrary category.

If the collection is directed, its direct limit is the union .

In the direct limit, this gives a projection P in .

For some categories C the direct limit used to define the stalk may not exist.

Therefore the direct limit collapses to yield S as the stalk.