What's the difference between
- morphisms between products in $\mathbf{Set}$ ($A_1\times A_2\rightarrow B_1\times B_2$), and
- morphisms between objects in $\mathbf{Set}\times\mathbf{Set}$ ($(A_1, A_2)\rightarrow (B_1, B_2)$)?
Here is my own answer -- can someone verify that this is correct?
Products in Set
Given sets $A_1, A_2, B_1, B_2 : \mathbf{Set}$, we have cartesian products
$$A_1\times A_2 : \mathbf{Set}$$ $$B_1\times B_2 : \mathbf{Set},$$
which are sets containing pairs. The morphisms
$$f : \text{Hom}_\mathbf{Set}(A_1 \times A_2, B_1 \times B_2)$$
between those sets are the two-argument functions of the form
$$f(a_1, a_2) = (\dots a_1 \dots a_2 \dots, \dots a_1 \dots a_2 \dots).$$
We can also refer to each component using angle-bracket notation:
$$\langle f_1, f_2\rangle : \text{Hom}_\mathbf{Set}(A_1 \times A_2, B_1 \times B_2)$$
with
$$f_1:\text{Hom}_\mathbf{Set}(A_1 \times A_2, B_1)$$ $$f_2:\text{Hom}_\mathbf{Set}(A_1 \times A_2, B_2)$$
Set x Set
By contrast, if we take the product category
$$\mathbf{Set}\times\mathbf{Set}:\mathbf{Cat},$$
we obtain the category of pairs of sets. Given $A_1, A_2, B_1, B_2 : \mathbf{Set}$, we have objects
$$(A_1, A_2):\mathbf{Set}\times\mathbf{Set},$$ $$(B_1, B_2):\mathbf{Set}\times\mathbf{Set}.$$
But unlike the earlier case, the morphisms between those two objects are pairs of functions
$$(f_1, f_2) : \text{Hom}_{\mathbf{Set}\times\mathbf{Set}}((A_1, A_2), (B_1, B_2))$$
where
$$f_1: \text{Hom}_{\mathbf{Set}}(A_1, B_1)$$ $$f_2: \text{Hom}_{\mathbf{Set}}(A_2, B_2)$$
of the form
$$f_1(a_1)=\dots a_1\dots$$ $$f_2(a_2)=\dots a_2\dots.$$
So $f_1$ cannot depend on $a_2$ and vice versa.
This looks right.
Essentially what you're saying is that every $f=(f_1,f_2) : (A_1,A_2) \to (B_1,B_2)$ in $\mathbf{Set} \times \mathbf{Set}$ induces a function $f_1 \times f_2 : A_1 \times A_2 \to B_1 \times B_2$ in $\mathbf{Set}$, where $f_1 \times f_2 = \langle f_1 \circ \pi_{A_1}, f_2 \circ \pi_{A_2} \rangle$, but not every function $A_1 \times A_2 \to B_1 \times B_2$ is of this form.
Or even more concisely, the product functor $\mathbf{Set} \times \mathbf{Set} \to \mathbf{Set}$ is not full.