 Solution

For a set $S$, let ${\iota }_s:1\to S$ select $s$. Given any family $f_s:1\to X$, define $f:S\to X$ by $f(s)=f_s(\ast )$. Then $f{\iota }_s=f_s$, and these equations determine every value of $f$, so it is unique.

This is precisely the universal property of a coproduct in a category, givingThe empty set gives the empty coproduct in a category, namely the initial object of the Category of sets.