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I'm trying to understand the categorical definition of a product, which describes them in terms of existence of a unique morphism that makes such-and-such a diagram commute. I don't really feel I've totally understood the motivation for this definition: in particular, why must that morphism be unique? What's the consequence of omitting the requirement for uniqueness in, say, Set?

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If you omit uniqueness, then any set admitting a surjective map to the usual cartesian product would satisfy the "universal" property. – Zhen Lin Jul 10 '12 at 13:02

This is a question which you will be able to answer yourself after some experience ... anyway:

The cartesian product $X \times Y$ of two sets $X,Y$ has the property: Every element of $X \times Y$ has a representation $(x,y)$ with unique elements $x \in X$ and $y \in Y$. This is the important and characteristic property of ordered pairs. In other words, if $*$ denotes the one-point set: For every two morphisms $x : * \to X$ and $y : * \to Y$ there is a unique morphism $(x,y) : * \to X \times Y$ such that $p_X \circ (x,y) = x$ and $p_Y \circ (x,y) = y$. But since we can do everything pointwise, the same holds for arbitrary sets instead of $*$: For every two morphisms $x : T \to X$ and $y : T \to Y$ (which you may think of families of elements in $X$ resp. $Y$, also called $T$-valued points in the setting of functorial algebraic geometry), there is a unique morphism $(x,y) : T \to X \times Y$ such that $p_X \circ (x,y) = x$ and $p_Y \circ (x,y) = y$. Once you have understood this in detail, this motivates the general definition of a product diagram in a category. After all, these appear everywhere in mathematics (products of groups, vector spaces, $\sigma$-algebras, topological spaces, etc.). Of course, the uniqueness statement is essential. Otherwise, the product won't be unique and in the case of sets you will get many more objects instead of the usual cartesian product. In general, an object $T$ satisfies the universal property of $X \times Y$ without the uniqueness requirement iff there is a retraction $T \to X \times Y$. So for example in the category of sets every set larger than the cartesian product will qualify.

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Well, a set-map $f:X\to A\times B$ should be uniquely determined by its components $f_A:X\to A$ and $f_B:X\to B$, and conversely any two functions $X\to A$ and $X\to B$ should combine to a map $X\to A\times B$.

This is basically a tautology in terms of the ususal construction of the set-product: writing $f(x)=(a_x,b_x)$ yields the functions $x\mapsto a_x$ and $x\mapsto b_x$.

In other words $f\mapsto (f_A,f_B)$ should be a bijection $$\text{Hom}(X,A\times B)\to \text{Hom}(X,A)\times \text{Hom}(X,B).$$

The uniqueness (resp. existence) part corresponds to injectivity (resp. surjectivity) of this map.

If you drop the uniqueness part, you will get many products. For example $A\times B\times Z$, for any set $Z$, will then be a product (with the projections to $A,B$). Indeed, the map

$$\text{Hom}(X,A\times B\times Z)\to \text{Hom}(X,A)\times \text{Hom}(X,B)$$

(sending a map to its $A,B$-projections) is surjective but highly non-injective: we are free to choose the $Z$-component.

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