How are functions $F: X \to Y^n$ constructed from $n$ functions $f_i: X \to Y$

Question

Given $n$ functions $$f_i : X \to Y$$ one can construct $$F^\prime: X^n \to Y^n$$ by taking a product over them. What about if I want to construct a function $$F: X \to Y^n$$ what mathematical operation achieves this?

Example in $\mathbb{R}$: $$f_0(x) = x$$ $$f_1(x) = x^2$$ $$F(x) = (x, x^2)$$ It is not a product, because a product would be $$F^\prime(x, y) = (x, y^2)$$ but I do not know what it is.

Answer 1

The fundamental property of product sets is

if $X,Y_1,Y_2,\dots,Y_n$ are sets and $f_i\colon X\to Y_i$ ($i=1,2,\dots,n$) are maps, then there exists a unique map $F\colon X\to Y_1\times Y_2\times\dots\times Y_n$ such that $\pi_i\circ F=f_i$, where $\pi_i\colon Y_1\times Y_2\times\dots\times Y_n\to Y_i$ is the projection map (that selects the $i$-th coordinate).

Of course the map $F$ is defined by $F(x)=(f_1(x),f_2(x),\dots,f_n(x))$.

In your case, $Y_1=Y_2=\dots=Y_n=Y$.

How's it called? It may be called the product map, why not?

Note that this is much more natural to construct than your $F'$.