Uniqueness of Representing Objects
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Theorem
Let $C$ be a locally small category.
Let $\mathbf{Set}$ be the category of sets.
Let $F : \mathbf C \to \mathbf{Set}$ be a covariant functor.
Let $(A, \eta)$ and $(B, \xi)$ be representations of $F$.
Then there exists a unique isomorphism $f : A \to B$ such that $\eta \circ h^f = \xi$, where:
- $h^f$ is the precomposition natural transformation
- $\circ$ denotes vertical composition of natural transformations
Proof
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