Coimage

In algebra, the coimage of a homomorphism

[math]\displaystyle{ f : A \rightarrow B }[/math]

is the quotient

[math]\displaystyle{ \text{coim} f = A/\ker(f) }[/math]

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 [math]\displaystyle{ f : X \rightarrow Y }[/math], then a coimage of [math]\displaystyle{ f }[/math] (if it exists) is an epimorphism [math]\displaystyle{ c : X \rightarrow C }[/math] such that

1. there is a map [math]\displaystyle{ f_c : C \rightarrow Y }[/math] with [math]\displaystyle{ f =f_c \circ c }[/math],
2. for any epimorphism [math]\displaystyle{ z : X \rightarrow Z }[/math] for which there is a map [math]\displaystyle{ f_z : Z \rightarrow Y }[/math] with [math]\displaystyle{ f =f_z \circ z }[/math], there is a unique map [math]\displaystyle{ h : Z \rightarrow C }[/math] such that both [math]\displaystyle{ c =h \circ z }[/math] and [math]\displaystyle{ f_z =f_c \circ h }[/math]

## See also

* Quotient object
* Cokernel

## References

* Mitchell, Barry (1965). Theory of categories. Pure and applied mathematics. 17. Academic Press. ISBN 978-0-124-99250-4.

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