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# Universal property of the empty set

Theorem : The empty set is the only set satisfying the universal property of the empty set.

Proof : The universal property of the empty set is the following : A set $X$ satisfies this universal property if, for every set $A$, there is a unique function from $X$ to $A$.

The proof is done in two steps. First, the empty set satisfies this property. Intuitively, this is because the only function with the empty set as its domain is the empty set itself, and the empty set is unique in most set theories. A function can be defined as a subset of the Cartesian product of the domain with the range obeying some properties, and the Cartesian product with the empty set is empty, therefore only the empty set can be such a function. The empty function is therefore our unique function to any set.

Secondly, any set obeying this property will also be the empty set. If a non-empty set $X$ obeys this property, then there is a unique map $f : \varnothing \to X$, and a unique map $g : X \to \varnothing$. As usual, we also have the identity map $\mathrm{Id}: \varnothing \to \varnothing$. Since the empty set obeys the universal property of the empty set, including with itself, $\mathrm{Id}$ is the unique function to itself.

The composition $g \circ f$, being a map from $\varnothing$ to itself, must therefore also be the identity, as that is the unique map. $f$ is therefore the inverse of $g$.