Definition:Unique up to Unique Morphism

Definition
Let $\mathbf C$ be a category.

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

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

Definition 2
The class $S$ is unique up to unique morphism 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 Morphism

Weaker properties

 * Definition:Unique up to Isomorphism
 * Definition:Unique up to Unique Isomorphism, see Unique up to Unique Morphism implies Unique up to Unique Isomorphism