Let $\mathcal{A}\to\textrm{Sets}$ be a covariant functor, $X, X'\in\operatorname{Obj}(\mathcal{A})$ objects, and $(X, \Phi), (X', \Phi')$ be representations of $F$, i.e. $\Phi: F\to\operatorname{Hom}_{\mathcal{A}}(X, \cdot)$ and $\Phi': F\to\operatorname{Hom}_{\mathcal{A}}(X', \cdot)$ are natural equivalences. Show that there is a unique isomorphism $f: X\to X'$ such that for any object $Y\in\operatorname{Obj}(\mathcal{A})$: $$\Phi_Y = \operatorname{Hom}_{\mathcal{A}}(\cdot, Y)(f)\circ \Phi_Y'$$
What I did so far:
$\Phi'\circ \Phi^{-1}: \operatorname{Hom}_{\mathcal{A}}(X, \cdot)\to\operatorname{Hom}_{\mathcal{A}}(X', \cdot)$
$\Phi\circ {\Phi'}^{-1}: \operatorname{Hom}_{\mathcal{A}}(X', \cdot)\to\operatorname{Hom}_{\mathcal{A}}(X, \cdot)$
are both (isomorphic) natural transformations. The Lemma of Yoneda states the bijectivity of the following mappings:
$$v: \operatorname{Nat}(\operatorname{Hom}_{\mathcal{A}}(X, \cdot), \operatorname{Hom}_{\mathcal{A}}(X', \cdot))\to\operatorname{Hom}_{\mathcal{A}}(X, X')$$ $$\phi\mapsto\phi_X(\textrm{Id}_X)$$ and $$v': \operatorname{Nat}(\operatorname{Hom}_{\mathcal{A}}(X', \cdot), \operatorname{Hom}_{\mathcal{A}}(X, \cdot))\to\operatorname{Hom}_{\mathcal{A}}(X', X)$$ $$\phi\mapsto\phi_{X'}(\textrm{Id}_{X'})$$
Now I could guess the morhpism that I search for is $f := v'(\Phi\circ {\Phi'}^{-1})$. In order to show that $f$ is an isomorphism, a good candidate for its inverse could be $\tilde{f} := v(\Phi'\circ \Phi^{-1})$.
So first I need to show that indeed $f\circ\tilde{f}=\textrm{Id}_X$. Afterwards I would like to show $$\forall Y\in\operatorname{Obj}(\mathcal{A}): \Phi_Y = \operatorname{Hom}_{\mathcal{A}}(\cdot, Y)(f)\circ \Phi_Y'$$ Finally I still need to show the uniqueness of $f$. So far I haven't really used the Lemma of Yoneda (only as a hint how to construct $f$).
I don't know whether this is the correct way. Any help is appreciated, especially hints that are easy to understand rather than complete abstract solutions.