Coimage

In algebra, the coimage of a homomorphism

is the quotient

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