Examples of categories

Categorically-encoded monoid

A monoid $M$ defines a category $𝐁M$ with a single object. $𝐁M$'s morphisms are exactly the objects of $M$. Composition of morphisms is given by multiplication.

Why isn't a category with category trivial? See Equality of morphisms: it is important to understand that an endomorphism (a morphism from an object to itself) is not necessarily an identity.

Category of small categories

Categories are fully-fledged mathematical objects in their own right, and can, amazingly, be modeled within the language of category theory itself. Unfortunately, to avoid Russel's paradox, we tend to only consider the category of small categories. That said, the category of small categories features objects of small categories, and morphisms of functors.

Preorder as a category

Given an arbitrary preorder, a category arises defined by the preorder relation (and the set it is defined upon), by reinterpreting reflexivity and transitivity as identity and composition, respectively.