HomSet with a single element -- empty function?

Question

I am currently reading Spivak's "Category Theory for the Sciences", and on page 16 the following definition is given:

Let $X$ and $Y$ be sets. We write $Hom_{Set}(X, Y)$ to denote the set of functions $X \rightarrow Y$.

Exercise 2.1.2.13.b. is to "Find a set $B$ such that for all sets $X$ there is exactly one element in $Hom_{Set}(B, X)$."

And the provided answer is $B=\emptyset$.

Is the single function the "empty function"?

Answer 1

Yes, it is indeed the empty function.${}{}{}{}$. In this case, the empty set is said to be initial.