In an abelian category, every map $f:B \to C$ factors as $B \xrightarrow{e} \text{im}(f) \xrightarrow{m} C$

In an abelian category, every map $f:B \to C$ factors as $B \xrightarrow{e} \text{im}(f) \xrightarrow{m} C$ with $m = \ker(\text{coker}f)$ monic and $e$ epi.

How can $\text{im}(f)$ be defined in an abelian category?

• The answer is in these notes. Get the most recent version and look in the section on abelian categories in the chapter on category theory: math.stanford.edu/~vakil/216blog – user4571 Dec 21 '15 at 21:31
• Thanks @Patrick, found it in the notes, but how do they identify $\text{im} f$ with an object? – BananaCats Category Theory App Dec 21 '15 at 21:38
• The image in an abelian category for a morphism $f:X\rightarrow Y$ is a morphism $\operatorname{im}(f)\rightarrow Y$, right? So just take the object named $\operatorname{im}(f)$. – user4571 Dec 21 '15 at 22:01

The image can't really be identified with an object: for example, the very fact every arrow factors as the coimage followed by the image means that image and coimage objects are isomorphic, so from looking at the object alone you cannot tell "which role it's playing". Usually, by slight abuse of notation, one identifies $\mathrm{Im}f$ with the subobject of $Y$ that it represents as a monomorphism.