# Coimage

In algebra, the coimage of a homomorphism

f: AB

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 : XY, then a coimage of f (if it exists) is an epimorphism c : XC such that

1. there is a map fc : CY with f = fcc,
2. for any epimorphism z : XZ for which there is a map fz : ZY with f = fzz, there is a unique map π : ZC such that both c = πz and fz = fcπ.

### Other articles related to "coimage, coimages":

Pre-abelian Category - Elementary Properties
... The existence of both kernels and cokernels gives a notion of image and coimage ... is, the image is the kernel of the cokernel, and the coimage is the cokernel of the kernel ... In many common situations, such as the category of sets, where images and coimages exist, their objects are isomorphic ...