FinSet

FinOrd is the category whose objects are all finite ordinal numbers and whose morphisms are all functions between them.

FinOrd is a full subcategory of FinSet as by the standard definition, suggested by John von Neumann, each ordinal is the well-ordered set of all smaller ordinals.

Unlike Set and FinSet, FinOrd is a small category.

As in Set, in FinSet the categorical product of two objects A and B is given by the cartesian product A × B, the categorical sum is given by the disjoint union A + B, and the exponential object BA is given by the set of all functions with domain A and codomain B.

The subobject classifier in FinSet and FinOrd is the same as in Set.