Uniqueness of Representing Objects

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