3. This composition is associative: h o (g o f) = (h o g) o f.
4. Each object A has an identity morphism id_A:A->A associated with it. This is the identity under composition, shown by the equations
id__B o f = f = f o id__A.
In general, the morphisms between two objects need notform a set (to avoid problems with Russell's paradox). An example of a category is the collection of sets where the objects are sets and the morphisms are functions.
Sometimes the composition ring is omitted. The use of capitals for objects and lower case letters for morphisms is widespread but not universal. Variables which refer to categories themselves are usually written in a scriptfont.