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.

Definition 1
The class $S$ is unique up to unique isomorphism for all objects $s,t \in S$ there is a unique isomorphism from $s$ to $t$.

Definition 2
The class $S$ is unique up to unique isomorphism for all objects $s,t \in S$ there is a unique morphism from $s$ to $t$.

Definition 3
The class $S$ is unique up to unique isomorphism for all objects $s,t \in S$ there is a unique morphism from $s$ to $t$, and it is an isomorphism.

Also see

 * Equivalence of Definitions of Unique up to Unique Isomorphism
 * Definition:Unique up to Isomorphism