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 f : X → Z and g : Y → Z with a common codomain. The pullback is written

Usually the morphisms f and g are omitted from the notation, and then the pullback is written

The pullback comes equipped with two natural morphisms P → X and P → Y. The pullback of two morphisms f and g need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, X ×_Z Y may intuitively be thought of as consisting of pairs of elements (x, y) with x in X, y in Y, and f(x)  =  g(y). 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.

Explicitly, a pullback of the morphisms ${\displaystyle f}$ and ${\displaystyle g}$ consists of an object ${\displaystyle P}$ and two morphisms ${\displaystyle p_{1}:P\rightarrow X}$ and ${\displaystyle p_{2}:P\rightarrow Y}$ for which the diagram

commutes. Moreover, the pullback (P, p₁, p₂) must be universal with respect to this diagram. That is, for any other such triple (Q, q₁, q₂) where q₁ : Q → X and q₂ : Q → Y are morphisms with f q₁ = g q₂, there must exist a unique u : Q → P 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 (A, a₁, a₂) and (B, b₁, b₂) of the same cospan X → Z ← Y, there is a unique isomorphism between A and B respecting the pullback structure.