Discrete category

A discrete category is a category whose only morphisms are the identity morphisms.

Given a set 𝐴, we get a discrete category 𝒞, in which the objects are the elements of 𝐴 and the morphisms are the identity morphisms, one for each 𝑥 ∈ 𝐴, which are uniquely determined by the identity axiom. A discrete category is so determined by its objects, which correspond exactly to its identity morphisms.