Uniqueness of morphism in definition of category theory product (etc) 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?

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.
