The function from the $\{\}$ to an any other set? In the $\mathcal{SET}$ - category of sets and maps between them - there is an initial object - the $\{\}$. It means that there is unique map from the $\{\}$ to an any other set (object of $\mathcal{SET}$).
Not sure I understand it completely. A map $\{\} \mapsto S \in Obj(\mathcal{SET})$ must take some $e \in \{\}$ to a some other $s \in S$. However, there is no such $e$ by the definition of the empty set. Thus such function does exist, however why is it unique? As long as particular $S$ contains more that one element, there are many maps of $\{\} \mapsto S$ kind. Thus the only way to prove that all those maps are essentially the same is to compare them to each other. In order to run comparsion, one has to kind of "compute" those functions - i.e. to assign some $e \in \{\}$ to each and ensure that outputs $s_0, s_1, ..., s_n \in S$ are the same. Otherwise, how can you determine that abovementioned functions are equal bypassing their evluation?

Think of it this way: we say two functions $f$ and $g$ on the same domain are different if there exists an $x$ in that domain so that $f(x) \neq g(x)$. In the case where the domain is empty, there obviously cannot be such an $x$, so $f$ and $g$ are automatically not different.
More formally, a function is formally defined as a set of ordered pairs $(e, f(e))$, where $e$ is taken from the domain. If the domain is empty, no such pairs can exist - so any function with domain $\{\}$ must itself be the empty set. And two "copies" of the empty set are automatically equal.
