Definition:Unique up to Unique Morphism

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 morphism if and only if 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 if and only if 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