Definition 0.1 \bigbreak A morphism in a category is an isomorphism when there exists an inverse morphism of in , denoted by Failed to parse (unknown function "\inv"): {\displaystyle \inv f: B \to A} , such that Failed to parse (unknown function "\inv"): {\displaystyle f \circ \inv f =id_A = 1_A: A \to A} .

One also writes: , expressing the fact that the object A is isomorphic with object B under the isomorphism .

Note also that an isomorphism is both a monomorphism and an epimorphism; moreover, an isomorphism is both a section and a retraction. However, an isomorphism is not the same as an equivalence relation.