morphism

monomorphism

A morphism ƒ:A→B is called a monomorphism if, for every object C and all morphisms g,h:C→A, the following holds.

ƒ∘g=ƒ∘h→g=h

epimorphism

A morphism ƒ:A→B is called an epimorphism if, for every object C and all morphisms g,h:B→C, the following holds.

g∘ƒ=h∘ƒ→g=h

bimorphism

A morphism ƒ:A→B is called a bimorphism if it is both a monomorphism and an epimorphism.

section

A morphism ƒ:A→B is called a section if there exists a morphism g:B→A such that g∘ƒ=(id_A).

retraction

A morphism ƒ:A→B is called a retraction if there exists a morphism g:B→A such that ƒ∘g=(id_B).

isomorphism

A morphism ƒ:A→B is called an isomorphism if there exists a morphism g:B→A such that g∘ƒ=(id_A) and ƒ∘g=(id_B).