TITLE: Why is the pullback not just a cartesian product (modulo a relation)?
QUESTION [5 upvotes]: Why is the pullback not just defined as in the case of the category of sets
http://en.wikipedia.org/wiki/Pullback_%28category_theory%29
Perhaps there are some issues with categories that are not small, but what about small categories? My first thought was: perhaps all small categories do not have products. Do they? If not, why doesn't the cartesian product just do the trick?

REPLY [5 votes]: Your intuition is right in the sense that pullbacks can be constructed from products and equalizers, see relation between pullbacks and other categorical limits, proposition 2. Products are generalizations of Cartesian products of sets. Equalizers are generalizations of specific subsets; I do not understand what is “modulo a relation”, but I feel equalizers are what you need. :)