Definition:Unique up to Isomorphism

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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 isomorphism if and only if for all objects $s,t \in S$ there is a isomorphism from $s$ to $t$.

Also see

Stronger properties