Let $\mathcal{A}$ be an abelian category. For a morphism $f:A\rightarrow B$, its image is usually defined in the following way: a map $i:I\rightarrow B$ is called an image of $f$ if it is a kernel of a cokernel of $f$.
I was trying to make an equivalent definition for the image of $f$ similar to the usual definitions of kernel and cokernels. I arrived to the following: a map $i:I\rightarrow B$ is called an image of $f$ if it is a monic satisfying the following two conditions:
- There exists a map $g:A\rightarrow I$ such that $ig=f$;
- If $j:J\rightarrow B$ is another map such that there exists $h:A\rightarrow J$ with $jh=f$ then there exits a unique map $k:I\rightarrow J$ such that $jk=i$.
I was trying to see if the condition for $i$ to be a monic is unnecessary but I coudn't proved it. So the question is, is a map $i:I\rightarrow B$ satisfying conditions 1 and 2 a monic?, or how I should modify 1 or 2 in order to eliminate the condition for $i$ to be a monic?.
Maybe it is better to put this in the category $R-mod$ with $R$ an associative ring. I would appreciate your suggestions in both cases.