Discussion
The pullback is similar to the product, but not the same. One may obtain the product by "forgetting" that the morphisms f and g exist, and forgetting that the object Z exists. One is then left with a discrete category containing only the two objects X and 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.
The categorical dual of a pullback is a called a pushout. Remarks opposite to the above apply: the pushout is a coproduct with additional structure.
Read more about this topic: Pullback (category Theory)
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