Coimage
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Coimage

In algebra, the coimage of a homomorphism

${\displaystyle f:A\rightarrow B}$

is the quotient

${\displaystyle {\text{coim}}f=A/\ker(f)}$

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

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