Possible Duplicate:
Category with zero morphisms
Given two objects $X,Y$ in a category, can there be more than one zero morphism $X\rightarrow Y$?
Possible Duplicate:
Category with zero morphisms
Given two objects $X,Y$ in a category, can there be more than one zero morphism $X\rightarrow Y$?
The definition of a zero morphism as a morphism that is both constant and coconstant is not a very useful one. Zero morphisms make much more sense in a category with a zero object – since in that case they can be defined to be the unique morphism that factors through the zero object. Under that definition, they are unique.
In the absence of a zero object, stupid things can happen. For example, consider the category $\{ \bullet \rightrightarrows \bullet \}$ with exactly two parallel non-identity arrows. It is essentially vacuous that both of the non-identity arrows are constant and coconstant – but they are distinct by hypothesis.