coimage

In algebra, the coimage of a homomorphism

:$f\; :\; A\; \backslash rightarrow\; B$

is the quotient

:$\backslash text\{coim\}\; f\; =\; A/\backslash ker(f)$

of the domain by the 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 $f\; :\; X\; \backslash rightarrow\; Y$, then a coimage of $f$ (if it exists) is an epimorphism $c\; :\; X\; \backslash rightarrow\; C$ such that

1. there is a map $f\_c\; :\; C\; \backslash rightarrow\; Y$ with $f\; =f\_c\; \backslash circ\; c$,
2. for any epimorphism $z\; :\; X\; \backslash rightarrow\; Z$ for which there is a map $f\_z\; :\; Z\; \backslash rightarrow\; Y$ with $f\; =f\_z\; \backslash circ\; z$, there is a unique map $h\; :\; Z\; \backslash rightarrow\; C$ such that both $c\; =h\; \backslash circ\; z$ and $f\_z\; =f\_c\; \backslash circ\; h$