I'm quoting from the Stacks Project:
Definition 4.3.6. A contravariant functor $C \to \text{Sets}$ is said to be representable if it is isomorphic to the functor of points $h_u$ for some object $u$ of $C$.
Let $C$ be a category and let $F: C^{op} \to \text{Sets}$ be a representable functor. Choose an object $u$ of $C$ and an isomorphism $s : h_u \to F$. The Yoneda lemma guarantees that the pair $(u,s)$ is unique up to unique isomorphism. The object $u$ is called an object representing $F$.
I am having trouble with the bolded part. I understand how to prove the Yoneda lemma as seen here in my answer.
So how do we go from $\text{Nat}(h_u, F) \simeq F(u)$ to $(u,s)$ is unique up to unique isomorphism? Please guide me as someone new to Yoneda lemma application.
Attempt
We have that $s : F \simeq h_u$ is a natural isomorphism of functors. Now $u$ is uniquely determined from $h_u$ and thus $s$ clearly. If $s' : F \simeq h_v$ then $s' \circ \theta = s$ for some natural isomorphism $\theta$ I suppose. And also $u \simeq v$. I guess I need to prove that now.
Well do $\theta = s'^{-1} s$ then clearly that isomorphism is unique. We can do that since they compose: $h_u \simeq F \simeq h_v$. Now $u \simeq v$ anyone?