Initial and terminal objects

Initial and terminal objects are not required to exist in a given category.

For complete categories there is an existence theorem for initial objects.

Specifically, a (locally small) complete category C has an initial object if and only if there exist a set I (not a proper class) and an I-indexed family (Ki) of objects of C such that for any object X of C, there is at least one morphism Ki → X for some i ∈ I. Terminal objects in a category C may also be defined as limits of the unique empty diagram 0 → C. Since the empty category is vacuously a discrete category, a terminal object can be thought of as an empty product (a product is indeed the limit of the discrete diagram {Xi}, in general).

Initial and terminal objects may also be characterized in terms of universal properties and adjoint functors.

Let 1 be the discrete category with a single object (denoted by •), and let U : C → 1 be the unique (constant) functor to 1.