I'm following Emily Riehl's book Category theory in context. In Theorem 2.2.4 (Yoneda lemma) it is stated: For any functor $F:\mathscr{C}\to\textbf{Set}$ whose domain $\mathscr{C}$ is a locally small category and any object $c\in\mathscr{C}$, there is a bijection
$$\text{Hom}(\mathscr{C}(c,-),F)\cong Fc$$
that associates the natural transformation $\alpha:\mathscr{C}(c,-)\Rightarrow F$ with components $(\alpha_d)$ to the element $\alpha_c(1_c)\in Fc$
For the proof, a natural transformation $\Psi(x):\mathscr{C}(c,-)\Rightarrow F$ is defined for every $x\in Fc$, specifying its components $\Psi(x)_d(f)=(Ff)(x)$ for any $f\in\mathscr{C}(c,d)$
I tried to prove the theorem head on starting with the injectivity of the asignment $\alpha\mapsto\alpha_c(1_c)$ but dind't get any far: why would $\alpha_c(1_c)=\beta_c(1_c)$ imply that $\alpha=\beta$ as natural transformations? Maybe $\alpha_c=\beta_c$ can be proved for that specific component but how are the other components affected?
Also, is there any rephrasing of Yoneda lemma in the 2-category $\textbf{Cat}$?
And last (not really importante to the question) is the notation $\text{Nat}(F,G)$ accepted and usual for the collection of natural transformations between the functors $F$ and $G$?
Thanks in advance
Best Answer
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.