Category of sets, $\mathrm{I}_{\emptyset}$

The identity map on the empty set is the empty map.

For every set $X$, there is an empty map $­∅ → X$. Depending on your set-theoretic implementation of the definition of a function, the empty map $∅ → X$ either

• coincides with $­∅$ itself (when viewed as a mere relation – that is, as a subset of $∅ × X = ∅$), or
• is the tuple $(∅,∅,X)$ (when viewed as a relation with a domain $∅$ and a codomain $X$), or
• is something else – but I honestly haven’t heard of any third set-theoretic definition of a map.

It’s the only map $∅ → X$, as there is only one relation between $∅­$ and $X$, as there is only one subset of $∅ × X$, namely $∅$ itself. This shows that the empty map $­∅ → ∅$ is an identity and that $∅$ is an initial object in the category of sets.

The same as any other identity function on a set: for $x\in\emptyset$, it's defined by $I_{\emptyset}(x)=x$. It just turns out the condition "$x\in\emptyset$" is vacuous in this case.