Category of small categories ( $Cat$ )

Objects are small categories and morphisms are functors. Its products pair objects and arrows componentwise. Its exponential objects are functor categories, as shown by the currying adjunction for small categories.

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Indiscrete category

The indiscrete category on a set has exactly one morphism between each ordered pair of objects. Every morphism is invertible. A functor into it is determined by its object assignment.

Discrete category

The discrete category on a set has that set as its objects and only identity morphisms. Functors from it are exactly assignments of objects in the target category.

Free category on independent arrows ( $L (S) = ∐_{s \in S} [1]$ )

Take two distinct objects and one nonidentity arrow for each element of

S

, without arrows between different copies. A functor from this category to

C

is exactly a choice of one arbitrary arrow of

C

for each element of

S

. Consequently this construction is left adjoint to the functor taking the set of all arrows of a small category. Its free object has three arrows per generator, including the two identity arrows.

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Category of small categories by

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The **category of small categories**, often denoted as **Cat**, is a mathematical category in category theory where the objects are small categories (categories that have a hom-set for every pair of objects that is a set, not a proper class) and the morphisms are functors between these categories. ### Key Elements: 1. **Objects**: The objects of **Cat** are **small categories**.

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