What are commonly called categories are actually Set-categories, in the terminology of \(\mathcal{V}\) categories.
There are many important categories:
Top - topological spaces (neighborhood)
Grph - graphs (connection)
Meas - measure spaces (amount)
Mon - monoids (action)
Grp - groups (reversible action, symmetry)
Cat - categories (action in context, structure)
The category of sets, denoted Set
Objects are the collection of all sets.
Morphisms are set-functions
Composition is function composition (satsifies associativity), identities are the identity functions (satisfies identity constraints).
Closely related is the subcategory FinSet of finite sets with morphisms being set-functions.
Any category \(\mathcal{C}\) induces another category, \(\mathcal{C}^{op}\) defined as the same objects but all arrows reversed.