Then a pullback of and (in ) is given by the preimage together with the inclusion of the preimage in and the restriction of to . Because of this example, in a general category the pullback of a morphism and a monomorphism can be thought of as the "preimage" under of the subobject specified by . Similarly, pullbacks of two monomorphisms can be thought of as the "intersection" of the two subobjects. Least common multiple Consider the multiplicative monoid of positive integers as a category with one object. In this category, the pullback of two positive integers and is just the pair , where the numerators are both the least common multiple of and . The same pair is also the pushout. Properties In any category with a terminal object , the pullback is just the ordinary product . Monomorphisms are stable under pullback: if the arrow in the diagram is monic, then so is the arrow . Similarly, if is monic, then so is . Isomorphisms are also stable, and hence, for example, for any map (where the implied map is the identity). In an abelian category all pullbacks exist, and they preserve kernels, in the following sense: if is a pullback diagram, then the induced morphism is an isomorphism, and so is the induced morphism . Every pullback diagram thus gives rise to a commutative diagram of the following form, where all rows and columns are exact: Furthermore, in an abelian category, if is an epimorphism, then so is its pullback , and symmetrically: if is an epimorphism, then so is its pullback . In these situations, the pullback square is also a pushout square. There is a natural isomorphism (A×CB)×B D ≅ A×CD. Explicitly, this means: if maps f : A → C, g : B → C and h : D → B are given and the pullback of f and g is given by r : P → A and s : P → B, and the pullback of s and h is given by t : Q → P and u : Q → D , then the pullback of f and gh is given by rt : Q → A and u : Q → D. Graphically this means that two pullback squares, placed side by side and sharing one morphism, form a larger pullback square when ignoring the inner shared morphism. Any