I need to prove that in an abelian category, the co-kernel of a 0-morphism is an isomorphism.
Let $A\rightarrow B$ be a $0$-morphism and $f:B\rightarrow C$ its cokernel. Then by the universal property of the the cokernel, I can show that there is a unique morphism $g:C\rightarrow B$ such that $g\circ f =id_B$. But I don't know how to prove that $f\circ g=id_C$.
I appreciate any help.