zero object as cokernel for an epimorphism
1. Proposition
Let 
be a pointed category and 
an epimorphism.
Then the cokernel is the zero object
2. Proof
Suppose there exists a morphism 
such that
commutes.
Then we conclude since the zero morphism factors through 
, that
and hence
Thus
commutes.
Uniqueness follows from the uniqueness of the zero morphism.