Definition:Unique up to Unique Isomorphism

Definition
Let $\mathbf C$ be a category.

Let $S \subseteq \operatorname{Ob}(\mathbf C)$ be a subclass of its objects.

The class $S$ is unique up to unique isomorphism for all objects $s,t \in S$ there is a unique isomorphism from $s$ to $t$.

Weaker properties

 * Definition:Unique up to Isomorphism

Stronger properties

 * Definition:Unique up to Unique Morphism, as shown at Unique up to Unique Morphism implies Unique up to Unique Isomorphism