Coimage

In algebra, the coimage of a homomorphism

f: A → B

is the quotient

coim f = A/ker f

of domain and kernel. The coimage is canonically isomorphic to the image by the first isomorphism theorem, when that theorem applies.

More generally, in category theory, the coimage of a morphism is the dual notion of the image of a morphism. If f : X → Y, then a coimage of f (if it exists) is an epimorphism c : X → C such that

1. there is a map f_c : C → Y with f = f_cc,
2. for any epimorphism z : X → Z for which there is a map f_z : Z → Y with f = f_zz, there is a unique map π : Z → C such that both c = πz and f_z = f_cπ.