Objects that are isomorphic to a terminal object are terminal themselves, and so the isomorphism turns to be unique. I am looking for an example of a limit or colimit (the vertex of the universal cone will be denoted $C$), such that there exists an object $C'$ (which maybe could be $C$ itself) which is isomorphic to $C$ but not uniquely so. I have tried to look this up but could not find any reference where this comes up.
Thank you.
This will happen more or less whenever you have an object with a non-trivial automorphism.
For example, in most abelian groups the automorphism $a\mapsto-a$ is non-trivial. If $G$ is such an abelian group (say $\mathbb{Z}$ under addition if you want a concrete example) then the product $G\times G$ has an automorphism which is different from the identity. (Actually it has at least three, because you can negate either component or both.)