I am studying Vakil's notes, Section 1.6.3, on the concept of abelian categories.
A kernel of a morphism $f: B\rightarrow C$ is a map $i: A\rightarrow B$ such that $f\circ i=0$, and that is universal with respect to this property. Diagramatically:
A cokernel is defined dually by reversing the arrows.
The image of a morphism $f: A\rightarrow B$ is defined as $\text{im}(f)=\ker(\text{coker (f)})$.
It is mentioned as a fact that
For a morphism $f: A\rightarrow B$, $A\rightarrow \text{im}(f)$ is an epimorphism. I will show an equivalent statement: $f: A\rightarrow B$ is an epimorphism if $\text{im}(f)=B$.
Here is my effort so far to show it.
We have the following diagram:
$e: B\rightarrow C$ is the cokernel of $f$, $i: B\rightarrow B$ is the kernel of the cokernel, hence the image of $f$. To show $f$ is an epimorphism, suppose that $\phi\circ f=0$ for some $\phi: B\rightarrow D$. We need to show that $\phi=0$.
We see that $\alpha\circ e\circ i=\phi\circ i=0$. My question is, when we say $\text{im}(f)=B$, does it implicitly imply that $\text{im}(f)$ is the identity map $\text{id}_B: B\rightarrow B$? If that is true, then I have $\phi=0$. Then the proof is done. If it is not, I don't know how to continue to show $\phi=0$.
Thank you for your help!