TITLE: What's actually the definition of the product of morphisms?
QUESTION [12 upvotes]: I have heard different versions of it, in wikipedia, it says that:


The unique morphism f is called the product of morphisms f1 and f2 and
  is denoted < f1, f2 >.

But in here, it says:
The product morphism of f and f′, denoted f×f′, is the unique morphism making the following diagram commute:

I am wondering are these two definition equivalent? If yes, why? If not, which is the correct definition?

REPLY [8 votes]: These are not equivalent definitions and they are defining two different morphisms, both of which are sometimes called the "product" of two morphisms.  So, both are "correct" and the term "product" is ambiguous and you'll have to figure out which one is meant from context.
In practice, there is usually little chance of confusion, especially since the first construction is only defined when the two morphisms have the same domain.  Personally, though, I would recommend calling the first definition the "tupling" of $f_1$ and $f_2$ rather than the "product" and reserve the term "product" for the second definition.