Editing Pullback (category theory) (section)

==Universal property==
Explicitly, a pullback of the morphisms <math>f</math> and <math>g</math> consists of an [[Object (category theory)|object]] <math>P</math> and two morphisms <math>p_1:P\rightarrow X</math> and <math>p_2:P\rightarrow Y</math> for which the diagram

:[[File:Categorical pullback.svg|125px|class=skin-invert]]

[[Commutative diagram|commutes]]. Moreover, the pullback {{math|(''P'', ''p''<sub>1</sub>, ''p''<sub>2</sub>)}} must be [[universal property|universal]] with respect to this diagram.<ref>Mitchell, p. 9</ref> That is, for any other such triple {{math|(''Q'', ''q''<sub>1</sub>, ''q''<sub>2</sub>)}} where {{math|''q''<sub>1</sub> : ''Q''&nbsp;→&nbsp;''X''}} and {{math|''q''<sub>2</sub> : ''Q''&nbsp;→&nbsp;''Y''}} are morphisms with {{math|''f'' ''q''<sub>1</sub>&nbsp;{{=}}&nbsp;''g'' ''q''<sub>2</sub>}}, there must exist a unique {{math|''u''&nbsp;:&nbsp;''Q''&nbsp;→&nbsp;''P''}} such that 

:<math>p_1 \circ u=q_1, \qquad p_2\circ u=q_2.</math>
This situation is illustrated in the following commutative diagram.
:[[File:Categorical pullback (expanded).svg|225px|class=skin-invert]]

As with all universal constructions, a pullback, if it exists, is unique up to [[isomorphism]].  In fact, given two pullbacks {{math|(''A'', ''a''<sub>1</sub>, ''a''<sub>2</sub>)}} and {{math|(''B'', ''b''<sub>1</sub>, ''b''<sub>2</sub>)}} of the same [[cospan]]  {{math|''X''&nbsp;→&nbsp;''Z''&nbsp;←&nbsp;''Y''}}, there is a unique isomorphism between {{mvar|A}} and {{mvar|B}} respecting the pullback structure.