Initial and terminal objects

An object $i$ in a category $C$ is called initial if for every object $c\in C$ there exists a unique arrow $i\to c$; in other words, the hom-sets ${\mathrm{Hom}}_C(i,c)$ are all singleton sets.

Dually, an object $t$ is called terminal if for every object $c\in C$ there exists a unique arrow $c\to t$, i.e., the hom-sets ${\mathrm{Hom}}_C(c,t)$ are all singleton sets.

1. Show that an initial (respectively, terminal) object, if it exists, is unique up to unique isomorphism.
2. Show that in $\mathbf{S}\mathbf{e}\mathbf{t}$, the empty set is initial and the singleton set is terminal.
3. Show that in $\mathbf{A}\mathbf{b}$, the trivial group $0$ is both initial and terminal. (Such an object is called a null or zero object.)
4. Show that $\mathbf{F}\mathbf{i}\mathbf{e}\mathbf{l}\mathbf{d}$ contains neither an initial nor a terminal object.