Editing Pullback (category theory) (section)

==Pullback and product==
The pullback is similar to the [[product (category theory)|product]], but not the same.  One may obtain the product by "forgetting" that the morphisms {{mvar|f}} and {{mvar|g}} exist, and forgetting that the object {{mvar|Z}} exists. One is then left with a [[discrete category]] containing only the two objects {{mvar|X}} and {{mvar|Y}}, 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" {{mvar|Z}}, {{mvar|f}}, and {{mvar|g}}, one can also "trivialize" them by specializing {{mvar|Z}} to be the [[terminal object]] (assuming it exists). {{mvar|f}} and {{mvar|g}} are then uniquely determined and thus carry no information, and the pullback of this cospan can be seen to be the product of {{mvar|X}} and {{mvar|Y}}.