Editing Pushout (category theory) (section)

==Properties==
*Whenever the pushout ''A''&thinsp;⊔<sub>''C''</sub> ''B'' exists, then ''B''&thinsp;⊔<sub>''C''</sub> ''A'' exists as well and there is a natural isomorphism ''A''&thinsp;⊔<sub>''C''</sub> ''B'' &cong; ''B''&thinsp;⊔<sub>''C''</sub> ''A''.
*In an [[abelian category]] all pushouts exist, and they preserve [[cokernel]]s in the following sense: if (''P'', ''i''<sub>1</sub>, ''i''<sub>2</sub>) is the pushout of ''f'' : ''Z'' &rarr; ''X'' and ''g'' : ''Z'' &rarr; ''Y'', then the natural map coker(''f'') &rarr; coker(''i''<sub>2</sub>) is an isomorphism, and so is the natural map coker(''g'') &rarr; coker(''i''<sub>1</sub>).
*There is a natural isomorphism (''A'' ⊔<sub>''C''</sub> ''B'')&thinsp;⊔<sub>''B''</sub> ''D'' &cong; ''A'' ⊔<sub>''C''</sub> ''D''. Explicitly, this means: 
** if maps ''f'' : ''C'' &rarr; ''A'', ''g'' : ''C'' &rarr; ''B'' and ''h'' : ''B'' &rarr; ''D'' are given and 
** the pushout of ''f'' and ''g'' is given by ''i'' : ''A'' &rarr; ''P'' and ''j'' : ''B'' &rarr; ''P'', and
** the pushout of ''j'' and ''h'' is given by  ''k'' : ''P'' &rarr; ''Q'' and ''l'' : ''D'' &rarr; ''Q'', 
** then the pushout of ''f'' and ''hg'' is given by ''ki'' : ''A'' &rarr; ''Q'' and  ''l'' : ''D'' &rarr; ''Q''. 
:Graphically this means that two pushout squares, placed side by side and sharing one morphism, form a larger pushout square when ignoring the inner shared morphism.