I am trying learn category theory from a programmer's perspective. Thus the following may be plain wrong. I want to know if my proof below is OK:
Show that the terminal object is unique upto a unique isomorphism
"Unique up to a unique isomorphism" implies the following: Unique up to unique isomorphism means that there's only one isomorphism relating the two isomorphic objects.
The terminal object is the object with one and only one morphism coming to it from any object in the category. Or alternatively, $T$ is terminal if for every object $X$ in $C$ there exists a single morphism $X \to T$.
So if $T_1$ and $T_2$ are two terminal objects, we need to prove that there is ONLY ONE isomorphism that relates $T_1$ and $T_2$.
By definition, "$T$ is terminal if for every object $X$ in $C$ there exists a single morphism $X \to T$."
Hence there should be a morphism from $T_1$ to $T_2$ and a morphism from $T_2$ to $T_1$ (apart from identity morphisms on $T_1$ and $T_2$).
There cannot be more than one morphism between $T_1$ and $T_2$.
Hence $T_1$ and $T_2$ has a unique up to a unique isomorphism.