We know that $\alpha_c(1_c)=\beta_c(1_c)$ and we want to prove that if $x\in \mathscr C$ is any object and $f\in \mathscr{C}(c, x)$, then
$$\alpha_x(f)=\beta_x(f)$$
Draw a commutative diagram that says that $\alpha$ is a natural transformation:
$$
\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}
\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex}
\newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}}
\begin{array}{c}
\mathscr{C}(c, c) & \ra{f_*} & \mathscr{C}(c,x) \\
\da{\alpha_c} & & \da{\alpha_x}\\
Fc & \ra{Ff} & Fx
\end{array}$$
Now for $1_c\in\mathscr{C}(c,c)$ we have
$(Ff)(\alpha_c(1_c))=\alpha_x(f_*(1_c))=\alpha_x(f)$. Drawing a similar diagram for $\beta$ we get exactly what we need $$\beta_x(f)=(Ff)(\beta_c(1_c))=(Ff)(\alpha_c(1_c))=\alpha_x(f)$$
I don't know how to express Yoneda lemma in 2-categorical terms, but there is a 2-categorical generalization of it, which you can look up in J. Hedman's 2-Categories and Yoneda lemma or nLab.
Nat$(F, G)$ is indeed popular notation for the set of natural transformations between two functors – see already mentioned 2-Categories and Yoneda lemma, J. Rotman's Homological algebra or Wikipedia.
There is an interesting variation of the Yoneda lemma at play here.
Note that a transformation with components $\phi_{X,Y}\colon C(X,Y)\to D(FX,FY)$ given by $\phi_{X,Y}:f\mapsto Ff$ is natural if and only if $F$ is a functor. Moreover, $F$ is full, resp. faithful, resp. fully faithful, if and only if the components are surjections, resp. injections, resp. bijections
The variation is then this: any natural transformation $\alpha\colon C(X,Y)\to D(FX,FY)$ is of the form $D(\beta,FY)\circ\phi=D(FX,\beta)\circ\phi$ for a natural transformation $\beta$ from $F$ to itself with components $\beta_X=\alpha_{X,X}(\mathrm{id}_X)$.
Indeed, naturality in $Y$ of $\alpha\colon C(X,Y)\to D(FX,FY)$ implies $\alpha_{X,Y}(f)=\alpha_{X,Y}\circ C(X,f)(\mathrm{id_X})=D(FX,Ff)\circ\alpha_{X,X}(\mathrm{id}_X)=Ff\circ\beta_X$ where $\beta_X=\alpha_{X,X}(\mathrm{id_X})\in D(FX,FY)$, while naturaliry in $X$ implies $\alpha_{X,Y}(f)=\beta_Y\circ Ff$. In particular, $Ff\circ\beta_X=\alpha_{X,Y}(f)=\beta_Y\circ Ff$, asserts exactly that $\beta$ is a natural transformation from $F$ to itself such that $D(\beta,FY)\circ\phi=\alpha=D(FX,\beta)\circ\phi$.
We now claim that if $\alpha_{X,X}$ are surjective, then $\beta$ is a natural isomorphism from $F$ to itself, whence $D(FX,\beta)$ and $D(\beta,FY)$ are natural bijections from $D(FX,FY)$ to $D(FX,FY)$ by which $\alpha$ and $\phi$ are related. In particular, $\alpha$ a natural bijection implies $\phi$ is a natural bijection, e.g. (YL-wtih-naturality) implies (YET).
Indeed, if $\alpha_{X,X}$ are surjective, then $\mathrm{id}_{FX}=Fs\circ j_X=j_X\circ Ft$ for some $s,t\in C(X,X)$, whence $Fs=Ft=\beta_X^{-1}$ are the unique two-sided inverses of $\beta_X$, from which follows that the natural transformation $\beta$ has an inverse $\beta^{-1}$.
Best Answer
When we say a bijection is natural, we mean it is natural in the sense of a natural transformation between functors. This almost always means that the square we get by changing the inputs to our bijection will commute, and with time you'll start to intuitively know what the square is and what naturality means.
For us, we have an isomorphism from $\text{Hom}(yc,F) \cong Fc$, where $y$ is the (covariant) yoneda embedding. If you like, this means we have two functors from $C \to \mathsf{Set}$, namely $\text{Hom}(y-,F)$ and $F$. Then "naturality in $c$" is saying that these two functors are naturally isomorphic.
Recall an isomorphism in $\mathsf{Set}$ is a bijection. So the fact that $\text{Hom}(yc,F) \cong Fc$ for each $c$ individually is already saying something interesting. But the fact that this isomorphism is natural means that if $f : c \to d$ is an arrow in $C$, then we get a commutative square
Intuitively, "naturality in $c$" means that these isomorphisms play nicely when we move between objects of $C$. If you have a programming background, you might see this as a kind of polymorphic isomorphism, in the sense that the same definition works uniformly for every object of $C$. See here for more, if this sounds like it might help.
With this in mind, can you guess what it means that these isomorphisms are also natural in $F$? Again, it will mean that two functors are naturally isomorphic, but now these functors will be $[C,\text{Set}] \to \mathsf{Set}$. You should try to draw the square yourself to get familiar working out what naturality means (most authors will leave it to you to figure out for yourself), but I'll include an answer under the fold:
As an aside, in this post we checked naturality in each variable separately. But you could also think of functors in two variables $\text{Hom}(y-,-) : C \times [C,\mathsf{Set}] \to \mathsf{Set}$ and $\text{ap} : C \times [C, \mathsf{Set}] \to \mathsf{Set}$, where $\text{ap}(c,F) = Fc$. If you're familiar with the subtletlies around separate and joint continuity, you might worry that naturality in $c$ and $F$ separately is different from naturality from the pair $(c,F)$.
Thankfully, this is not the case, and the notions of "separate" and "joint" naturality in multiple variables agree. See here, for instance.
I hope this helps ^_^