Lets take a category of sets and functions between them. So there are (infinitely?) many terminal objects. As far as I understand, it's no so hard to prove that any two terminal objects are isomorphic to each other.
Concept of being "unique up to isomorphism" still seems a bit unclear for me, so I'm wondering wether it is technically correct to say that those singletons-terminal objects are "unique up to isomorphism". It feels like I can go further and say that those are unique up to unique isomorphism, because the definition a of terminal object implies that no other isomorphism can take place simply becase no more arrows in-between any two terminal objects can appear.