Why does the terminal set have exactly one element?

Question

I'm reading Lawvere/Rosebrugh's Sets for Mathematics.

He states the axiom of the terminal set:

AXIOM: TERMINAL SET

There is a set $1$ such that for any set $A$ there is exactly one mapping $A \to 1$. This unique mapping is given the same name $A$ as the set that is its domain.

And then he gives this definition:

Definition 1.3: An element of a set $A$ is any mapping whose codomain is $A$ and whose domain is $1$ (or abbreviated... $1 \stackrel{\alpha }{\longrightarrow} A$).

And then asks:

Why does $1$ itself have exactly one element according to this definition?

I answered the following:

Taking the axiom of the terminal set, which states that there is a unique mapping $A \to 1$ and taking the definition of mapping (for each $x\in A$ there is an $x\in B$), then every element of $A$ is connected to one element of $1$, and the only way to achieve a unique mapping in this respect is to have exactly one element in $A$.

Is it correct?

Answer 1

Not really. You shouldn’t be thinking in terms of the familiar notion of mapping at all. Just apply the axiom to the set $A=1$: there is a unique mapping $1\to 1$. Each element $\alpha$ of $1$ is a mapping $1\overset{\alpha}\longrightarrow 1$, and there is only one such mapping, so $1$ has a unique element.

Answer 2

To have two different elements in the set 1, you need two different mappings from 1 to the set 1 - that's the definition of an element. But there is a unique mapping from 1 to 1 by the terminal set axiom.