A seemingly simple fact about construction of maps proven categorically, i.e. by universal properties

Question

If I have four sets $A,B,C,D$ and two maps $f_1 : A \to C$ and $f_2 : B \to D$, it is easy to find a unique map $f : A\times B \to C\times D$, namely $$ f(a,b) := (f_1(a), f_2(b)). $$ But now I want to show this using the universal property of the direct produkt, and that we could construct mappings between such product in all categories. The universal property reads, that for each $f_1 : Y \to C, f_2 : Y \to D$ there exists a unique map $f : Y \to C \times D$ such that $$ f_1 = \pi_1 \circ f \qquad f_2 = \pi_2 \circ f. $$ So, to apply it in this case, I have to set $Y = A\times B$ in some way. Thinking in conrete sets, the maps $$ f_1'(a,b) = f_1(a) = (f_1\circ \pi_1)(a,b) \qquad \mbox{ and } \quad f_2'(a,b) = f_2(b) = (f_2 \circ \pi_2)(a,b) $$ would work. But to what universal property does this "ignoring" of the first or second component corresponds... I could not reduce it to the universal property of the direct product, because there I have mappings with domain an arbitrary object, but here I have mappings which have the product as domain, so everything is reversed... Any hints? Or maybe I did something wrong and this property of SETS could not be generalized to arbitrary categories and products (of two objects)...

Answer 1

Since $A\times B$ is a product, you have projection maps $\pi_1\colon A\times B\to A$ and $\pi_2\colon A\times B\to B$ as part of the definition of product (I use the same names as for the product $C\times D$, but the context makes clear which is meant). Now we have the maps $f_1\circ\pi_1\colon A\times B\to C$ and $f_2\circ \pi_2\colon A\times B\to D$, hence by the universal property for the product $C\times D$ a unique map $f\colon A\times B\to C\times D$ such that $\pi_1\circ f = f_1\circ \pi_1$ and $\pi_2\circ f = f_2\circ \pi_2$.

More generally, if $I$ is an index set and we have maps $f_i\colon A_i\to B_i$ for all $i\in I$, the same construction gives us a unique map $f\colon \prod_{i\in I} A_i\to \prod_{i\in I} B_i$ with the property $\pi_i\circ f=f_i\circ \pi_i$ for all $i\in I$.