Coimage

In algebra, the coimage of a homomorphism


 * $$f : A \rightarrow B$$

is the quotient


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

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 $$f : X \rightarrow Y$$, then a coimage of $$f$$  (if it exists) is an epimorphism $$c : X \rightarrow C$$ such that
 * 1) there is a map $$f_c : C \rightarrow Y $$ with $$ f =f_c \circ c $$,
 * 2) for any epimorphism $$z : X \rightarrow Z$$ for which there is a map $$f_z : Z \rightarrow Y $$ with $$ f =f_z \circ z $$, there is a unique map $$ h : Z \rightarrow C $$ such that both $$ c =h \circ z $$ and $$ f_z =f_c \circ h $$