Zero morphism

In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object.
Definitions.
Suppose C is a category, and "f" : "X" → "Y" is a morphism in C. The morphism "f" is called a constant morphism (or sometimes left zero morphism) if for any object "W" in C and any , "fg" = "fh". Dually, "f" is called a coconstant morphism (or sometimes right zero morphism) if for any object "Z" in C and any "g", "h" : "Y" → "Z", "gf" = "hf". A zero morphism is one that is both a constant morphism and a coconstant morphism.
A category with zero morphisms is one where, for every two objects "A" and "B" in C, there is a fixed morphism 0"AB" : "A" → "B", and this collection of morphisms is such that for all objects "X", "Y", "Z" in C and all morphisms "f" : "Y" → "Z", "g" : "X" → "Y", the following diagram commutes:
The morphisms 0"XY" necessarily are zero morphisms and form a compatible system of zero morphisms.
If C is a category with zero morphisms, then the collection of 0"XY" is unique.
This way of defining a "zero morphism" and the phrase "a category with zero morphisms" separately is unfortunate, but if each hom-set has a ″zero morphism", then the category "has zero morphisms".
Related concepts.
If C has a zero object 0, given two objects "X" and "Y" in C, there are canonical morphisms "f" : "X" → 0 and "g" : 0 → "Y". Then, "gf" is a zero morphism in MorC("X", "Y"). Thus, every category with a zero object is a category with zero morphisms given by the composition 0"XY" : "X" → 0 → "Y".
If a category has zero morphisms, then one can define the notions of kernel and cokernel for any morphism in that category.