Cats in context Lemma 2.4.7. I'm failing to understand  the argument behind Lemma 2.4.7. in page 86 from Emily Riehl's Category Theory in Context.
In particular along the following reasoning:

Here $\int F$ is the category of elements of the functor $F$ (which is defined shortly before in the chapter) and   $\Psi:Fc\to \operatorname{Hom}(C(c,-),F)$ is the bijection defined in the Yoneda Lemma proof.
From it's definition, $\Psi(x):C(c,-)\implies F$ is a natural transformation such that it's components satisfy
$$\Psi(x)_a (g)=Fg(x)$$
for each $g\in C(c,a)$ and where $x\in Fc$.
I was under the impression that what's being done here is deducing the construction of $\int F$ from the properties of the Yoneda bijections and embeddings (since one reads "The Yoneda lemma supplies an alternative definition of the category of elements of F." at the begining of the previous paragraph).
But when I read that 'a natural transformation is given by a morphism such that the diagram commutes' as in the picture, I'm led to believe that that's already part of the definition of $\int F $ taking place. Is the commutativity of the diagram immediate from the Yoneda results? Even if not, I failed to prove it on my own.
In either case, I'm in need of some clarification on this matter. What's the point of this section?
Thank you.
 A: Suppose you first start with a contravariant functor $\newcommand{\C}{\textsf{C}} F \colon \C^{\rm op} \to \textsf{Set}$, and you want to define a category $\int F$ whose objects are “elements” $x \in Fc$, where $c \in \C$.
The question is, what is the most natural way to define the morphisms?
Riehl says that we must first identify the objects with its images under the bijection of Yoneda’s lemma, that is, we identify $x \in Fc$ with the natural transformation $\Psi(x) \colon \C(-,c) \to F$ whose component at $d \in \C$ is the function
$$
\Psi(x)_d \colon \C(d,c) \to Fd, \quad f \mapsto Ff(x).
$$
Once we do this, Riehl define the morphisms as follows:
A morphism from $\Psi(x) \colon \C(-,c) \to F$ to $\Psi(x’) \colon \C(-,c’) \to F$ is a natural transformation $\eta \colon \C(-,c) \to \C(-,c’)$ that makes the following diagram commute.
$$
\require{AMScd}
\begin{CD}
\C(-,c) @>{\Psi(x)}>> F \\
@V{\eta}VV @| \\
\C(-,c’) @>>{\Psi(x’)}> F
\end{CD}
$$
But note that $\eta = f_*$ if $f := \eta_c(1_c)$, so we may think of $\eta$ as a morphism $f \colon c \to c’$ such that
$$
\require{AMScd}
\begin{CD}
\C(-,c) @>{\Psi(x)}>> F \\
@V{f_*}VV @| \\
\C(-,c’) @>>{\Psi(x’)}> F
\end{CD}
$$
commutes. What does that mean? Well, taking the component at $c$ on this diagram of natural transformations we get the diagram
$$
\require{AMScd}
\begin{CD}
\C(c,c) @>{\Psi(x)_c}>> Fc \\
@V{f_*}VV @| \\
\C(c,c’) @>>{\Psi(x’)_c}> Fc
\end{CD}
$$
and evaluating at the identity $1_c$ we obtain that
\begin{align}
x = 1_{Fc}(x) = F(1_c)(x) &= \Psi(x)_c(1_c) \\
&= \Psi(x’)_c(f_*(1_c)) = \Psi(x’)(f) = Ff(x’),
\end{align}
as desired.
The covariant case is handled similarly.
