Let $\text{Cat}$ be the category of small categories. I am interested in the Yoneda embedding
$$ Y : \text{Cat}^{op} \rightarrow [\text{Cat}, \text{Cat}]$$
$Y$ is a $2$-functor- it can be applied to categories (objects in $X$), functors ($1$-morphisms in $X$), and natural transformations ($2$-morphisms). $Y(C)$ is a functor (an object in $[\text{Cat}, \text{Cat}]$). $Y(C)(D) = [C, D]_{\text{Cat}}$. $Y(F)$ is a natural transformation (a $1$-morphism in $[\text{Cat}, \text{Cat}]$).
My question is about $Y(\eta)$ for a natural transformation $\eta : F \implies G$ ($2$-morphism in $C$). I can't seem to work out what the data of $Y(\eta)$ is – it should be a $2$-morphism in $[\text{Cat}, \text{Cat}]$.
Best Answer
It seems like your question is more about what the 2-morphisms in $\newcommand\Cat{\mathbf{Cat}}[\Cat,\Cat]\newcommand\C{\mathcal{C}}\newcommand\D{\mathcal{D}}$ are, rather than what the data of $Y(\eta)$ is specifically.
Let's do this a little more generally. Let $\C$, $\D$ be (strict) 2-categories. Then $[\C,\D]$ should also be a (strict) 2-category, and we want to understand the 0, 1, and 2-cells.
0-cells:
The objects are strict 2-functors, i.e., functors $F:\C\to \D$ which act on objects, morphisms, and 2-morphisms subject to compatibility criteria. More concretely, once we've decided where $F$ sends objects, then the maps on hom categories $$F_{X,Y} : \C(X,Y)\to \D(X,Y)$$ should all be functors, and moreover, $$ \require{AMScd} \begin{CD} \C(Y,Z)\times \C(X,Y) @>\circ_{\C,X,Y,Z}>>\C(X,Z)\\ @VF_{Y,Z}\times F_{X,Y}VV @VVF_{X,Z}V\\ \D(FY,FZ)\times \D(FX,FY) @>\circ_{\D,FX,FY,FZ}>>\D(FX,FZ)\\ \end{CD} $$ should strictly commute.
1-cells:
The morphisms are (strictly) natural families of 1-cells. I.e., given $F,G:\C\to \D$, a 1-cell from $F$ to $G$ is a family $T_X : FX\to GX$ of 1-cells in $\D$, subject to the requirement that the usual diagram commute strictly for each 1-cell $f:X\to Y$ in $\C$: $$ \begin{CD} FX @>Ff>> FY\\ @VT_X VV @VVT_Y V \\ GX @>Gf>> GY. \\ \end{CD} $$
2-cells:
Let $F,G :\C \to \D$ be 2-functors, $T,S : F\to G$ be 1-cells between them. A 2-cell $\alpha : T \to S$ is a natural family of 2-cells. More concretely, it is the choice for every $X\in C$ of a 2-cell in $\D$, $\alpha_X : T_X\to S_X$ natural in the sense that for every 1-cell of $\C$, $f:X\to Y$, we have that the following 2-cells from $G(f)\circ T_X = T_Y\circ F(f)$ to $G(f)\circ S_X = S_Y\circ F(f)$ are equal. The two cells are the whiskered composites $G(f).\alpha_X$ and $\alpha_Y.F(f)$.
Applying this to $\C=\D=\Cat$
Given a 2-cell $\eta : F\to G$ in $\Cat$, we need to produce for each category $C$ a 2-cell $Y(\eta)_C : Y(F)_C\to Y(G)_C$.
If $X$ and $Y$ are the categories such that $F,G:X\to Y$, then $Y(F)_C: [Y,C]\to [X,C]$ is the functor $-\circ F$, and similarly for $G$. Then $Y(\eta)_C$ should be the whiskered composite $-.\eta$.
In other words, for any functor $K:Y\to C$, for all $x\in X$, by definition, $\eta_X : FX\to GX$, so $K.\eta_X = K(\eta_X) : KFX\to KGX$ is a natural transformation.