Definition of a Category, Isomorphisms

Definition 4.1. A category C is a pair of classes (0,A4) satisfying certain properties. The class O is called the class of objects, and the class A4 is called the class o/morphisms. 

The class M is actually a disjoint union of sets Ad (a, b), for every pair a,b O, with a given composition rule 

This composition rule is required to satisfy the following axioms  

The following terminology corresponds to the intuitive notion of equality between objects  

Clearly, being isomorphic is an equivalence relation. First, the inverse of the isomorphism has an inverse (the isomorphism itself), hence is an isomorphism too. Second, a composition of two isomorphisms is also an isomorphism, since by associativity its inverse is simply the composition of the inverses of the two isomorphisms, taken in the reverse order. 

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