# Representability $h_C(X) \simeq F$ of functors determines rep. $X$ up to unique isomorphism (using Yoneda Lemma).

A functor $F \in C^{\wedge}$ from $C^{op}$ to $\text{Set}$ is representable if there is an isomoprhism $\varphi \in \text{Hom}_{C^{\wedge}}(h_C(X), F)$ for some $X \in C$, we can also write that as $h_C(X) \simeq F$.

I want to prove that $X$ would be determined up to unique isomorphism.

Proof. Suppose that $\varphi: h_C(X) \simeq F$, and $\phi : h_C(Y) \simeq F$, then $\phi \varphi^{-1} : h_C(X) \simeq h_C(Y) \in \text{Hom}_{C^{\wedge}}(h_C(X), h_C(Y)) \simeq h_C(X,Y)$.

How do I complete the proof?

Thanks @Anton

That got me this far:

We have a functor from $C^{op} \to \text{Set}$ given by $h_C(X)$ which is fully faithful. By functoriality, any isomorphism in $C$ is also one under $h_C(X) \circ \text{op}$. Since $h_C$ is fully faithful, we have that an isomorphism in $h_C(X, Y)$ corresponds to one in $\text{Hom}_{C^{\wedge}}(h_C(X), h_C(Y))$ under the Yoneda bijection, $\psi$. Let the "forward" Yoneda bijection be $\varphi$. Then $h_C(X) \simeq F, \ h_C(Y) \simeq F$ gives an isomorphism $h_C(X) \simeq h_C(Y): \theta$. Then $\varphi(\theta) : X \to Y$. We need to show that $\varphi(\theta)$ is unique (the only such isomorphism). Well if $\phi$ were another isomorphism, then $\phi^{-1} \varphi(\theta)$ is an automorphism of $X$. We know that $\text{Hom}_{C^{\wedge}}(h_C(X), h_C(X)) \simeq h_C(X,X)$

But still stuck.

• @EyesOnBud Your proof is finished, because the natural isomorphism $\theta$ admits a unique representing morphism $\phi(\theta)$ by Yoneda's lemma. You only need to prove that it is an isomorphism, which you already did: the inverse morphism represents the inverse natural transformation and the inverse laws hold by Yoneda's lemma. Aug 19, 2017 at 5:14
Since $\varphi$ is a bijection there can be only a unique $\theta$ such that $\varphi(\theta) \colon h_C(X) \to h_C(Y)$ is your natural isomorphism between the representable presheaves.
• @EyesOnBud you said it yourself "Let the "forward" Yoneda bijection be $\varphi$". Aug 20, 2017 at 11:44