How does the categorical definition of a product apply to the arithmetical product? If object $X$ and morphisms $p_1 : X \to X_1, p_2 : X \to X_2$ define the product of objects $X_1, X_2$ in category $C$, how do we map this construction to arithmetical multiplication? What are the objects and morphisms in $C$? What items can we identify with the objects and morphisms involved in this product?

The categorical definition of a product is _not_ intended to be a generalization of the arithmetic notion of a product. Rather, it is intended as a generalization of the notion of a Cartesian product of sets (or more complicated structures, like groups, topological spaces, etc.). Specifically, the Cartesian product $A\times B=\\{(a,b):a\in A,b\in B\\}$ of sets $A$ and $B$ together with the projection maps $p_1:A\times B\to A$ and $p_2:A\times B\to B$ given by $p_1(a,b)=a$ and $p_2(a,b)=b$ is a product in the category whose objects are sets and whose morphisms are functions.

Cartesian products of sets, of course, are themselves named after arithmetic products of numbers, since they are closely related. Specifically, if $A$ is a finite set with $a$ elements and $B$ is a finite set with $b$ elements, then the set $A\times B$ has $a\cdot b$ elements.