pullback

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