Category of small categories

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Definition

The category of small categories, sometimes denoted $\operatorname {Cat}$, is defined as follows:

• The objects of the category are small categories
• The morphisms of the category are functors: The morphisms between two small categories are defined as all the functors between those categories.
• The identity morphism is defined as the identity functor of a small category.
• The composition of morphisms is defined by composition of functors.

The category of small categories is a locally small category.