pushout

{{category theory}} Given a pair of arrows <math>f:X\rightarrow Y</math> and <math>g: X\rightarrow Z</math> with a common domain, ''X'', their '''pushout''' is a pair of arrows <math>u:Y\rightarrow P</math> and <math>v:Z\rightarrow P</math> with common codomain, ''P'', such that the equation <math> u\circ f = v\circ g</math> is satisfied, and for which there is the ''universal property'' that for any other object ''W'' for which there are also arrows <math>m: Y\rightarrow W</math>, <math>n: Z\rightarrow W</math>; there is a unique arrow <math>k: P\rightarrow W</math> such that <math> k\circ u = m</math> and <math>k\circ v = n</math>.