In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often written and comes equipped with two natural morphisms and . The pullback of two morphisms and need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, may intuitively be thought of as consisting of pairs of elements with in , in , and . For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative square. The dual concept of the pullback is the pushout. Universal property Explicitly, a pullback of the morphisms and consists of an object and two morphisms and for which the diagram commutes. Moreover, the pullback must be universal with respect to this diagram. That is, for any other such triple where and are morphisms with , there must exist a unique such that This situation is illustrated in the following commutative diagram. As with all universal constructions, a pullback, if it exists, is unique up to isomorphism. In fact, given two pullbacks and of the same cospan , there is a unique isomorphism between and respecting the pullback structure. Pullback and product The pullback is similar to the product, but not the same. One may obtain the product by "forgetting" that the morphisms and exist, and forgetting that the object exists. One is then left with a discrete category containing only the two objects and , and no arrows between them. This discrete category may be used as the index set to construct the ordinary binary product. Thus, the pullback can be thought of as the ordinary (Cartesian) product, but with additional structure. Instead of "forgetting" , , and , one can also "trivialize" them by specializing to be the terminal object (assuming it exists). and are then uniquely determined and thus carry no information, and the pullback of this cospan can be seen to be the product of and . Examples Commutative rings In the category of commutative rings (with identity), the pullback is called the fibered product. Let , , and be commutative rings (with identity) and and (identity preserving) ring homomorphisms. Then the