What is Pushout (category theory)?

In category theory, a branch of mathematics, a pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain. The pushout consists of an object P along with two morphisms X → P and Y → P that complete a commutative square with the two given morphisms f and g. In fact, the defining universal property of the pushout essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are P = X ⊔ Z Y {\displaystyle P=X\sqcup _{Z}Y} and P = X + Z Y {\displaystyle P=X+_{Z}Y}.

The pushout is the categorical dual of the pullback.