Why does the Category Theory Definition of $\mathbf{Set}$ Product Not Define a Subset of a Product? 1 Definition of Product
From the Wikipedia Definition of a Categorical Product (in the simple binary case):

2 Question
Now let us focus our attention on $\mathbf{Set}$.
$$
Y = \{1, 2 \} \\
X_1 = \{a, b \} \\
X_2 = \{c, d \} \\
$$
Then if $1 \mapsto a, c$ and $2 \mapsto b, d$ (under each respective $f_1$ and $f_2$), then wouldn't we have that $Y$ maps to $\{ (a, c), (b, d) \}$ rather than the full $X_1 \times X_2$?
If that's the case, then how does the Categorical Definition of a product square with the Set Theory definition (in the $\mathbf{Set}$)?

I think you're not being precise enough with the crucial question:

wouldn't we have that $Y$ maps to $\{ (a, c), (b, d) \}$

Well, certainly there does exist a function $g:Y\to \{(a,c),(b,d)\}$ such that $g(1)=(a,c)$ and $g(2)=(b,d)$. But this $g$ is not the product of $f_1$ and $f_2$, precisely because of the definition of a product: the codomain of $g$ is not (the Cartesian product of sets) $X_1\times X_2$.
(And yes, we know that the Cartesian product of sets is indeed the categorical product. It sounds like you don't doubt that part.)
