Editing Zero morphism (section)

==Definitions==
Suppose '''C''' is a [[Category (mathematics)|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 (category theory)|object]] ''W'' in '''C''' and any {{nowrap|''g'', ''h'' : ''W'' → ''X''}}, ''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<sub>''AB''</sub> : ''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:

[[Image:ZeroMorphism.png|center|160px]]

The morphisms 0<sub>''XY''</sub> 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<sub>''XY''</sub> is unique.<ref>{{cite web|url=https://math.stackexchange.com/q/189818 |title=Category with zero morphisms - Mathematics Stack Exchange |website=Math.stackexchange.com |date=2015-01-17 |access-date=2016-03-30}}</ref>

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 unique "zero morphism", then the category "has zero morphisms".