Sets and functions do form a category .

The objects of are sets and its arrows are functions.

For any set we identify with the identity function on , .

is indeed associative.

Each poset is a category.

The objects of each poset are its elements and we construct an arrow for every pair of objects iff .

Reflexivity () and transitivity () provide a construction for and , respectively.

Each monoid is a category.

The objects of each monoid are its elements and we construct for every pair of elements an arrow .

The existence of an identity element and associativity, as required by the monoid structure, immediately provide us with a construction for and associativity of .