Why is an empty set not a terminal object in categories $\mathsf{Top}$ and $\mathsf{Sets}$?

From Awodey: In any category $\mathsf{C}$,

an object $0$ is initial if for any object $C$ there is a unique morphism $0 \to C$,

an object $1$ is terminal if for any object $C$ there is a unique morphism $C \to 1$.

First I was puzzled by the fact that we can do mappings from a zero set to any other set in a category $\mathsf{Sets}$. But then after I accepted it conceptually, I do not understand why there is no unique morphism to an empty set from every object in a category $\mathsf{Sets}$. Is this because we cannot map a non-empty set to an empty set (since our mapping is total) although we can map an empty set to a non-empty set?

Please, note, I am familiar with the proposition saying: Initial (terminal) objects are unique up to isomorphism.

Solution 1:

Is this because we cannot map a non-empty set to an empty set (since our mapping is total) although we can map an empty set to a non-empty set?

Yep. A function has to take values on inputs. A function with non-empty domain but empty codomain can't take any values. (By contrast, a function with empty domain but non-empty codomain doesn't need to take values because it has no inputs.)

Solution 2:

I think you already answered your own question: Yes, this is because a mapping (in $\mathcal{Set}$) are defined to be total.